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In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.^[1]^[2]^[3] For example,

1200 = 2^{4} \times 3^{1} \times 5^{2} = 3 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 = 5 \times 2 \times 3 \times 2 \times 5 \times 2 \times 2 = \dots etc.

The theorem is stating two things: first, that 1200 can be represented as a product of primes, and second, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.

The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (e.g. 12 = 2 × 6 = 3 × 4).

History

Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the fundamental theorem. Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.

Applications

Canonical representation of a positive integer

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{k}^{α_{k}} = \prod_{i = 1}^{k} p_{i}^{α_{i}}

where Template:Nowrap beginp₁ < p₂ < ... < p_kTemplate:Nowrap end are primes and the α_i are positive integers.

This representation is called the canonical representation^[4] of n, or the standard form^[5]^[6] of n.

For example 999 = 3³×37, 1000 = 2³×5³, 1001 = 7×11×13

Note that factors p⁰ = 1 may be inserted without changing the value of n (e.g. 1000 = 2³×3⁰×5³).
In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers,

n = 2^{n_{2}} 3^{n_{3}} 5^{n_{5}} 7^{n_{7}} \dots = \prod p_{i}^{n_{p_{i}}} .

where a finite number of the n_i are positive integers, and the rest are zero. Allowing negative exponents provides a canonical form for positive rational numbers.

Arithmetic operations

This representation is convenient for expressions like these for the product, gcd, and lcm:

a \cdot b = 2^{a_{2} + b_{2}} 3^{a_{3} + b_{3}} 5^{a_{5} + b_{5}} 7^{a_{7} + b_{7}} \dots = \prod p_{i}^{a_{p_{i}} + b_{p_{i}}},

\gcd (a, b) = 2^{\min (a_{2}, b_{2})} 3^{\min (a_{3}, b_{3})} 5^{\min (a_{5}, b_{5})} 7^{\min (a_{7}, b_{7})} \dots = \prod p_{i}^{\min (a_{p_{i}}, b_{p_{i}})},

lcm (a, b) = 2^{\max (a_{2}, b_{2})} 3^{\max (a_{3}, b_{3})} 5^{\max (a_{5}, b_{5})} 7^{\max (a_{7}, b_{7})} \dots = \prod p_{i}^{\max (a_{p_{i}}, b_{p_{i}})} .

While expressions like these are of great theoretical importance their practical use is limited by our ability to factor numbers.

Arithmetical functions

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Many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers.

Proof

The proof uses Euclid's lemma (Elements VII, 30): if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b (or both). The article has proofs of the lemma.

Existence

By induction: assume it is true for all numbers between 1 and n. If n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and Template:Nowrap begin1 < a ≤ b < n.Template:Nowrap end By the induction hypothesis, Template:Nowrap begina = p₁p₂...p_nTemplate:Nowrap end and Template:Nowrap beginb = q₁q₂...q_mTemplate:Nowrap end are products of primes. But then Template:Nowrap beginn = ab = p₁p₂...p_nq₁q₂...q_mTemplate:Nowrap end is the product of primes. In the base case, 2 is a trivial product of primes.

Uniqueness

Assume that s > 1 is the product of prime numbers in two different ways:

\begin{aligned} s & = p_{1} p_{2} \dots p_{m} \\ = q_{1} q_{2} \dots q_{n} . \end{aligned}

We must show m = n and that the q_j are a rearrangement of the p_i.

By Euclid's lemma, p₁ must divide one of the q_j; relabeling the q_j if necessary, say that p₁ divides q₁. But q₁ is prime, so its only divisors are itself and 1. Therefore, p₁ = q₁, so that

\begin{aligned} \frac{s}{p_{1}} & = p_{2} \dots p_{m} \\ = q_{2} \dots q_{n} . \end{aligned}

Reasoning the same way, p₂ must equal one of the remaining q_j. Relabeling again if necessary, say p₂ = q₂. Then

\begin{aligned} \frac{s}{p_{1} p_{2}} & = p_{3} \dots p_{m} \\ = q_{3} \dots q_{n} . \end{aligned}

This can be done for all m of the p_i, showing that m ≤ n. If there were any q_j left over we would have

\begin{aligned} \frac{s}{p_{1} p_{2} \dots p_{m}} & = 1 \\ = q_{k} \dots q_{n}, \end{aligned}

which is impossible, since the product of numbers greater than 1 cannot equal 1. Therefore m = n and every q_j is a p_i.

Elementary proof of uniqueness

The fundamental theorem of arithmetic can also be proved without using Euclid's lemma, as follows:

Assume that s > 1 is the smallest positive integer which is the product of prime numbers in two different ways. If s were prime then it would factor uniquely as itself, so there must be at least two primes in each factorization of s:

\begin{aligned} s & = p_{1} p_{2} \dots p_{m} \\ = q_{1} q_{2} \dots q_{n} . \end{aligned}

If any p_i = q_j then, by cancellation, s/p_i = s/q_j would be a positive integer greater than 1 with two distinct factorizations. But s/p_i is smaller than s, meaning s would not actually be the smallest such integer. Therefore every p_i must be distinct from every q_j.

Without loss of generality, take p₁ < q₁ (if this is not already the case, switch the p and q designations.) Consider

t = (q_{1} - p_{1}) (q_{2} \dots q_{n}),

and note that 1 < q₂ ≤ t < s. Therefore t must have a unique prime factorization. By rearrangement we see,

\begin{aligned} t & = q_{1} (q_{2} \dots q_{n}) - p_{1} (q_{2} \dots q_{n}) \\ = s - p_{1} (q_{2} \dots q_{n}) \\ = p_{1} ((p_{2} \dots p_{m}) - (q_{2} \dots q_{n})) . \end{aligned}

Here u = ((p₂ ... p_m) - (q₂ ... q_n)) is positive, for if it were negative or zero then so would be its product with p₁, but that product equals t which is positive. So u is either 1 or factors into primes. In either case, t = p₁u yields a prime factorization of t, which we know to be unique, so p₁ appears in the prime factorization of t.

If (q₁ - p₁) equaled 1 then the prime factorization of t would be all q's, which would preclude p₁ from appearing. Thus (q₁ - p₁) is not 1, but is positive, so it factors into primes: (q₁ - p₁) = (r₁ ... r_h). This yields a prime factorization of

t = (r_{1} \dots r_{h}) (q_{2} \dots q_{n}),

which we know is unique. Now, p₁ appears in the prime factorization of t, and it is not equal to any q, so it must be one of the r's. That means p₁ is a factor of (q₁ - p₁), so there exists a positive integer k such that p₁k = (q₁ - p₁), and therefore

p_{1} (k + 1) = q_{1} .

But that means q₁ has a proper factorization, so it is not a prime number. This contradiction shows that s does not actually have two different prime factorizations. As a result, there is no smallest positive integer with multiple prime factorizations, hence all positive integers greater than 1 factor uniquely into primes.

Generalizations

The first generalization of the theorem is found in Gauss's second monograph (1832) on biquadratic reciprocity. This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now denoted by $ℤ [i] .$ He showed that this ring has the four units ±1 and ±i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes.^[7]

Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring $ℤ [ω]$ , where $ω = \frac{- 1 + \sqrt{- 3}}{2},$ $ω^{3} = 1$ is a cube root of unity. This is the ring of Eisenstein integers, and he proved it has the six units $\pm 1, \pm ω, \pm ω^{2}$ and that it has unique factorization.

However, it was also discovered that unique factorization does not always hold. An example is given by $ℤ [\sqrt{- 5}]$ . In this ring one has^[8]

6 = 2 \times 3 = (1 + \sqrt{- 5}) \times (1 - \sqrt{- 5}) .

Examples like this caused the notion of "prime" to be modified. In $ℤ [\sqrt{- 5}]$ it can be proven that if any of the factors above can be represented as a product, e.g. 2 = ab, then one of a or b must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; e.g. 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6. In algebraic number theory 2 is called irreducible (only divisible by itself or a unit) but not prime (if it divides a product it must divide one of the factors). Using these definitions it can be proven that in any ring a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers $ℤ$ every irreducible is prime". This is also true in $ℤ [i]$ and $ℤ [ω],$ but not in $ℤ [\sqrt{- 5}] .$

The rings where every irreducible is prime are called unique factorization domains. As the name indicates, the fundamental theorem of arithmetic is true in them. Important examples are Euclidean domains and principal ideal domains.

In 1843 Kummer introduced the concept of ideal number, which was developed further by Dedekind (1876) into the modern theory of ideals, special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains.

Notes

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References

The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n". Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n".

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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148; German translations are pp. 511–533 and 534–586 of the German edition of the Disquisitiones.

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

Here is my web site - cottagehillchurch.com
22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

Here is my web site - cottagehillchurch.com

External links

GCD and the Fundamental Theorem of Arithmetic at cut-the-knot.
PlanetMath: Proof of fundamental theorem of arithmetic
Fermat's Last Theorem Blog: Unique Factorization, a blog that covers the history of Fermat's Last Theorem from Diophantus of Alexandria to the proof by Andrew Wiles.
"Fundamental Theorem of Arithmetic" by Hector Zenil, Wolfram Demonstrations Project, 2007.
Template:Cite web

Template:Fundamental theorems Template:Divisor classes

de:Primfaktorzerlegung#Fundamentalsatz der Arithmetik

↑ Template:Harvtxt
↑ Template:Harvtxt
↑ Hardy & Wright, Thm 2
↑ Template:Harvtxt
↑ Template:Harvtxt
↑ Hardy & Wright § 1.2
↑ Gauss, BQ, §§ 31–34
↑ Hardy & Wright, § 14.6

[1] Template:Harvtxt

[2] Template:Harvtxt

[3] Hardy & Wright, Thm 2

[4] Template:Harvtxt

[5] Template:Harvtxt

[6] Hardy & Wright § 1.2

[7] Gauss, BQ, §§ 31–34

[8] Hardy & Wright, § 14.6

[1]

[2]

[3]

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[5]

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Fluorescence: Difference between revisions

Revision as of 17:47, 2 February 2014

Contents

History

Applications

Canonical representation of a positive integer

Arithmetic operations

Arithmetical functions

Proof

Existence

Uniqueness

Elementary proof of uniqueness

Generalizations

See also

Notes

References

External links

Navigation menu

@@ Line 1: / Line 1: @@
-== AzGxIk Brazil 6s 7D2rgA7u ==
+{{Distinguish|Fundamental theorem of algebra}}
-If you've tried Rhapsody in the past or you're a current subscriber, hit the comments and let me know what you think of the service. For a buck, I'm definitely [http://www.pickpm.com/pickss.php??p=14038-White-Blue-Shox-Conundrum-Online-Sale.html White Blue Shox Conundrum Online Sale] going to get my three months' worth. (If you see a different price, that means the page has defaulted back to a different seller because Click Go and Buy has run out of stock.) That is perhaps the best tablet deal I've seen to date, especially if you root that sucker to run Android 4.1. Read CNET's review of the HD+ if you want to learn more.<br><br>September poll, which found Obama leading Romney 60 percent to 38 percent among likely Massachusetts voters. That poll was done as Obama was seeing a bounce nationally after the Democratic National Convention. National polls released in mid September found Obama continuing to maintain an edge over Romney even as the race remained close. Romney was likely hurt by comments he made at a private fundraiser, in which he characterized 47 percent of Americans, who do not pay federal income taxes and who support Obama, as who feel entitled to government support. (Romney has since backed [http://www.mpif.org/partss.php??p=19734-buy-Nike-Air-Max-90-Black-Blue-for-men.html buy Nike Air Max 90 Black Blue for men] off from those comments.) On Friday, the website Real Clear Politics, which aggregates polling data, put Obama ahead of Romney nationally by three points. But Romney may win some support back after his strong performance in last week debate.<br><br>Gourcuff arrival at Stade Gerland is therefore the final piece, Aulas hopes, in a title winning puzzle.Already this term it been painfully obvious that the Rhone side were going to have issues breaking down stubborn defences. Home games against Monaco and Stade Brestois have yielded only one goal, and though that equated to four points, it hardly the dominant form that Lyon sides are used to.Admittedly an Achilles injury has prevented star forward and record signing Lisandro Lopez to feature in any form to date, and when the Argentine returns OL should boast a far more fearsome offensive line, despite the good work done by Bafetimbi Gomis in the opening month of the campaign.But Gourcuff will provide his side with another dimension from the midfield, even if his fee does look rather steep in comparison to Mesut Oezil, who switched from Werder Bremen to Real Madrid for just Worries have been correctly expressed over the playmaker recent form. <br><br>That poll pleased California Assemblyman Tom Ammiano, a San Francisco Democrat who introduced a [http://www.pickpm.com/pickss.php??p=41954-buy-cheap-Nike-Lunarglide-3-Cool-Grey-Black-Sonic-Yellow-Reflect4e-Silver-Shield-Shoes-for-women-sale.html buy cheap Nike Lunarglide 3 Cool Grey Black Sonic Yellow Reflect4e Silver Shield Shoes for women sale] bill in February to legalize marijuana in a manner similar to alcohol   taxing sales to adults while barring possession by anyone under 21. Ammiano hopes for a vote by early next year and contends the bill would generate up to $1.3 billion in revenue for his deficit plagued state.<br><br>Richard Nixon: Less than six months before he resigned amid the Watergate scandal, Nixon played piano at the Grand Ole Opry on its first night in Donelson after leaving the Ryman. He played "My Wild Irish Rose" in honor of his wife. Seigenthaler, who was backstage that night in 1974, said country music legend Roy Acuff, who had run for governor as a Republican in 1948, was "ecstatic."
+In [[number theory]], the '''fundamental theorem of arithmetic''', also called the '''unique factorization theorem''' or the '''unique-prime-factorization theorem''', states that every [[integer]] greater than 1 either is prime itself or is the product of [[prime number]]s, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.<ref>{{harvtxt|Long|1972|p=44}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=53}}</ref><ref>Hardy & Wright, Thm 2</ref> For example,
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- </ul>
-== dduqop World Cup Brazil 6s 7qBX6bCD ==
+: <math>
+= 2^4 \times 3^1 \times 5^2
+= 3 \times 2\times 2\times 2\times 2 \times 5 \times 5
+= 5 \times 2\times 3\times 2\times 5 \times 2 \times 2
+=\cdots\text { etc.}
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+\!
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+</math>
-  <ul>
+The theorem is stating two things: first, that 1200 ''can'' be represented as a product of primes, and second, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.
-   <li>[http://www.applesteps.com/Main_Page http://www.applesteps.com/Main_Page]</li>
+The requirement that the factors be prime is necessary: factorizations containing [[composite number]]s may not be unique (e.g. 12 = 2 × 6 = 3 × 4).
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+==History==
-   <li>[http://www.jdzchys.com/shownews.asp?id=50 http://www.jdzchys.com/shownews.asp?id=50]</li>
+Book VII, propositions 30 and 32 of [[Euclid]]'s [[Euclid's Elements|Elements]] is essentially the statement and proof of the fundamental theorem. Article 16 of [[Carl Friedrich Gauss|Gauss]]' ''[[Disquisitiones Arithmeticae]]'' is an early modern statement and proof employing [[modular arithmetic]].
- </ul>
+==Applications==
+===Canonical representation of a positive integer=== <!-- Redirect [[Canonical form]] links here -->
+Every positive integer ''n'' > 1 can be represented '''in exactly one way''' as a product of prime powers:
+:<math>
+n
+= p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}
+= \prod_{i=1}^{k}p_i^{\alpha_i}
+</math>
+where {{nowrap begin}}''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>k</sub>{{nowrap end}} are primes and the α<sub>''i''</sub> are positive integers.
+This representation is called the '''canonical representation'''<ref>{{harvtxt|Long|1972|p=45}}</ref> of ''n'', or the '''standard form'''<ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=55}}</ref><ref>Hardy & Wright § 1.2</ref> of ''n''.
+:For example 999 = 3<sup>3</sup>×37, 1000 = 2<sup>3</sup>×5<sup>3</sup>, 1001 = 7×11×13
+Note that factors ''p''<sup>0</sup> = 1 may be inserted without changing the value of ''n'' (e.g. 1000 = 2<sup>3</sup>×3<sup>0</sup>×5<sup>3</sup>).<br>In fact, any positive integer can be uniquely represented as an [[infinite product]] taken over all the positive prime numbers,
+:<math>
+n=2^{n_2}3^{n_3}5^{n_5}7^{n_7}\cdots=\prod p_i^{n_{p_i}}.
+</math>
+where a finite number of the ''n''<sub>''i''</sub> are positive integers, and the rest are zero. Allowing negative exponents provides a canonical form for positive [[rational number]]s.
+===Arithmetic operations===
+This representation is convenient for expressions like these for the product, [[greatest common divisor|gcd]], and [[least common multiple|lcm]]:
+:<math>
+a\cdot b
+=2^{a_2+b_2}\,3^{a_3+b_3}\,5^{a_5+b_5}\,7^{a_7+b_7}\cdots
+=\prod p_i^{a_{p_i}+b_{p_i}},
+</math>
+:<math>
+\gcd(a,b)
+=2^{\min(a_2,b_2)}\,3^{\min(a_3,b_3)}\,5^{\min(a_5,b_5)}\,7^{\min(a_7,b_7)}\cdots
+=\prod p_i^{\min(a_{p_i},b_{p_i})},
+</math>
+:<math>
+\operatorname{lcm}(a,b)
+=2^{\max(a_2,b_2)}\,3^{\max(a_3,b_3)}\,5^{\max(a_5,b_5)}\,7^{\max(a_7,b_7)}\cdots
+=\prod p_i^{\max(a_{p_i},b_{p_i})}.
+</math>
+While expressions like these are of great theoretical importance their practical use is limited by our ability to [[Integer factorization|factor]] numbers.
+===Arithmetical functions===
+{{Main|Arithmetic function}}
+Many arithmetical functions are defined using the canonical representation. In particular, the values of [[additive function|additive]] and [[multiplicative function|multiplicative]] functions are determined by their values on the powers of prime numbers.
+==Proof==
+The proof uses [[Euclid's lemma]] (''Elements'' VII, 30): if a prime ''p'' [[Divisor|divides]] the product of two [[natural number]]s ''a'' and ''b'', then ''p'' divides ''a'' or ''p'' divides ''b'' (or both). The article has proofs of the lemma.
+===Existence===
+By induction: assume it is true for all numbers between 1 and ''n''. If ''n'' is prime, there is nothing more to prove. Otherwise, there are integers ''a'' and ''b'',  where ''n'' = ''ab'' and {{nowrap begin}}1 < ''a'' ≤ ''b'' < ''n''.{{nowrap end}}
+By the induction hypothesis,
+{{nowrap begin}}''a'' = ''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''n''</sub>{{nowrap end}}
+and
+{{nowrap begin}}''b'' = ''q''<sub>1</sub>''q''<sub>2</sub>...''q''<sub>''m''</sub>{{nowrap end}} are products of primes. But then
+{{nowrap begin}}''n'' = ''ab'' = ''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''n''</sub>''q''<sub>1</sub>''q''<sub>2</sub>...''q''<sub>''m''</sub>{{nowrap end}} is the product of primes. In the base case, 2 is a trivial product of primes.
+===Uniqueness===
+Assume that ''s'' > 1 is the product of prime numbers in two different ways:
+:<math>
+\begin{align}
+s
+&=p_1 p_2 \cdots p_m \\
+&=q_1 q_2 \cdots q_n.
+\end{align}
+</math>
+We must show ''m'' = ''n'' and that the ''q''<sub>''j''</sub> are a rearrangement of the ''p''<sub>''i''</sub>.
+By [[Euclid's lemma]], ''p''<sub>1</sub>  must divide one of the ''q''<sub>''j''</sub>; relabeling the ''q''<sub>''j''</sub> if necessary, say that ''p''<sub>1</sub> divides ''q''<sub>1</sub>. But ''q''<sub>1</sub> is prime, so its only divisors are itself and 1. Therefore, ''p''<sub>1</sub> = ''q''<sub>1</sub>, so that
+:<math>
+\begin{align}
+\frac{s}{p_1}
+&=p_2 \cdots p_m \\
+&=q_2 \cdots q_n.
+\end{align}
+</math>
+Reasoning the same way, ''p''<sub>2</sub> must equal one of the remaining ''q''<sub>''j''</sub>. Relabeling again if necessary, say ''p''<sub>2</sub> = ''q''<sub>2</sub>. Then
+:<math>
+\begin{align}
+\frac{s}{p_1 p_2}
+&=p_3 \cdots p_m \\
+&=q_3 \cdots q_n.
+\end{align}
+</math>
+This can be done for all ''m'' of the ''p''<sub>''i''</sub>, showing that ''m'' ≤ ''n''. If there were any ''q''<sub>''j''</sub> left over we would have
+:<math>
+\begin{align}
+\frac{s}{p_1 p_2 \cdots p_m}
+&=1 \\
+&=q_k \cdots q_n,
+\end{align}
+</math>
+which is impossible, since the product of numbers greater than 1 cannot equal 1. Therefore ''m'' = ''n'' and every ''q''<sub>''j''</sub> is a ''p''<sub>''i''</sub>.
+===Elementary proof of uniqueness===
+The fundamental theorem of arithmetic can also be proved without using Euclid's lemma, as follows:
+Assume that ''s'' > 1 is the smallest positive integer which is the product of prime numbers in two different ways. If ''s'' were prime then it would factor uniquely as itself, so there must be at least two primes in each factorization of ''s'':
+:<math>
+\begin{align}
+s
+&=p_1 p_2 \cdots p_m \\
+&=q_1 q_2 \cdots q_n.
+\end{align}
+</math>
+If any ''p''<sub>''i''</sub> = ''q''<sub>''j''</sub> then, by cancellation, ''s''/''p''<sub>''i''</sub> = ''s''/''q''<sub>''j''</sub> would be a positive integer greater than 1 with two distinct factorizations. But ''s''/''p''<sub>''i''</sub> is smaller than ''s'', meaning ''s'' would not actually be the smallest such integer. Therefore every ''p''<sub>''i''</sub> must be distinct from every ''q''<sub>''j''</sub>.
+Without loss of generality, take ''p''<sub>1</sub> < ''q''<sub>1</sub> (if this is not already the case, switch the ''p'' and ''q'' designations.) Consider
+:<math>t = (q_1 - p_1)(q_2 \cdots q_n),</math>
+and note that 1 < ''q''<sub>2</sub> ≤ ''t'' < ''s''. Therefore ''t'' must have a unique prime factorization. By rearrangement we see,
+:<math>
+\begin{align}
+t
+&= q_1(q_2 \cdots q_n) - p_1(q_2 \cdots q_n) \\
+&= s - p_1(q_2 \cdots q_n) \\
+&= p_1((p_2 \cdots p_m) - (q_2 \cdots q_n)).
+\end{align}
+</math>
+Here ''u'' = ((''p''<sub>2</sub> ... ''p''<sub>''m''</sub>) - (''q''<sub>2</sub> ... ''q''<sub>''n''</sub>)) is positive, for if it were negative or zero then so would be its product with ''p''<sub>''1''</sub>, but that product equals ''t'' which is positive. So ''u'' is either 1 or factors into primes. In either case, ''t'' = ''p''<sub>1</sub>''u'' yields a prime factorization of ''t'', which we know to be unique, so ''p''<sub>1</sub> appears in the prime factorization of ''t''.
+If (''q''<sub>1</sub> - ''p''<sub>1</sub>) equaled 1 then the prime factorization of ''t'' would be all ''q'''s, which would preclude ''p''<sub>1</sub> from appearing. Thus  (''q''<sub>1</sub> - ''p''<sub>1</sub>) is not 1, but is positive, so it factors into primes: (''q''<sub>1</sub> - ''p''<sub>1</sub>) = (''r''<sub>1</sub> ... ''r''<sub>''h''</sub>). This yields a prime factorization of
+:<math>t = (r_1 \cdots r_h)(q_2 \cdots q_n),</math>
+which we know is unique. Now, ''p''<sub>1</sub> appears in the prime factorization of ''t'', and it is not equal to any ''q'', so it must be one of the ''r'''s. That means ''p''<sub>1</sub> is a factor of (''q''<sub>1</sub> - ''p''<sub>1</sub>), so there exists a positive integer ''k'' such that ''p''<sub>1</sub>''k'' = (''q''<sub>1</sub> - ''p''<sub>1</sub>), and therefore
+:<math>p_1(k+1) = q_1.</math>
+But that means ''q''<sub>1</sub> has a proper factorization, so it is not a prime number. This contradiction shows that ''s'' does not actually have two different prime factorizations. As a result, there is no smallest positive integer with multiple prime factorizations, hence all positive integers greater than 1 factor uniquely into primes.
+==Generalizations==
+The first generalization of the theorem is found in Gauss's second monograph (1832) on [[biquadratic reciprocity]]. This paper introduced what is now called the [[ring theory|ring]] of [[Gaussian integer]]s, the set of all [[complex number]]s ''a'' + ''bi'' where ''a'' and ''b'' are integers. It is now denoted by <math>\mathbb{Z}[i].</math> He showed that this ring has the four units ±1 and ±''i'', that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes.<ref>Gauss, BQ, §§ 31–34</ref>
+Similarly, in 1844 while working on [[cubic reciprocity]], [[Gotthold Eisenstein|Eisenstein]] introduced the ring <math>\mathbb{Z}[\omega]</math>, where   <math>\omega=\frac{-1+\sqrt{-3}}{2}, </math> &nbsp; <math>\omega^3=1 </math> is a cube [[root of unity]]. This is the ring of [[Eisenstein integer]]s, and he proved it has the six units <math>\pm 1, \pm\omega, \pm\omega^2</math> and that it has unique factorization.
+However, it was also discovered that unique factorization does not always hold. An example is given by <math>\mathbb{Z}[\sqrt{-5}]</math>. In this ring one has<ref>Hardy & Wright, § 14.6</ref>
+:<math>
+=
+\times 3=
+(1+\sqrt{-5})\times(1-\sqrt{-5}).
+</math>
+Examples like this caused the notion of "prime" to be modified. In <math>\mathbb{Z}[\sqrt{-5}]</math> it can be proven that if any of the factors above can be represented as a product, e.g. 2 = ''ab'', then one of ''a'' or ''b'' must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; e.g.
+divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6. In [[algebraic number theory]] 2 is called '''irreducible''' (only divisible by itself or a unit) but not '''prime''' (if it divides a product it must divide one of the factors). Using these definitions it can be proven that in any ring a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers <math>\mathbb{Z}</math> every irreducible is prime". This is also true in <math>\mathbb{Z}[i]</math> and <math>\mathbb{Z}[\omega],</math> but not in <math>\mathbb{Z}[\sqrt{-5}].</math>
+The rings where every irreducible is prime are called [[unique factorization domain]]s. As the name indicates, the fundamental theorem of arithmetic is true in them. Important examples are [[Euclidean domain]]s and [[principal ideal domain]]s.
+In 1843 [[Ernst Kummer|Kummer]] introduced the concept of [[ideal number]], which was developed further by [[Richard Dedekind|Dedekind]] (1876) into the modern theory of [[Ideal (ring theory)|ideals]], special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called [[Dedekind domain]]s.
+==See also==
+* [[Riemann zeta function#Euler product formula|Euler product formula]]
+* [[Integer factorization]]
+* [[Noetherian ring]]
+* [[Prime signature]]
+==Notes==
+{{reflist}}
+== References ==
+The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
+* {{citation
+  | last1 = Gauss  | first1 = Carl Friedrich
+  | last2 = Clarke | first2 = Arthur A. (translator into English)
+  | title = Disquisitiones Arithemeticae (Second, corrected edition)
+  | publisher = [[Springer Science+Business Media|Springer]]
+  | location = New York
+  | year = 1986
+  | isbn = 0387962549}}
+* {{citation
+  | last1 = Gauss  | first1 = Carl Friedrich
+  | last2 = Maser | first2 = H. (translator into German)
+  | title = Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
+  | publisher = Chelsea
+  | location = New York
+  | year = 1965
+  | isbn = 0-8284-0191-8}}
+The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § ''n''". Footnotes referencing the ''Disquisitiones Arithmeticae'' are of the form "Gauss, DA, Art. ''n''".
+* {{citation
+  | last1 = Gauss  | first1 = Carl Friedrich
+  | title = Theoria residuorum biquadraticorum, Commentatio prima
+  | publisher = Comment. Soc. regiae sci, Göttingen 6
+  | location = Göttingen
+  | year = 1828}}
+* {{citation
+  | last1 = Gauss  | first1 = Carl Friedrich
+  | title = Theoria residuorum biquadraticorum, Commentatio secunda
+  | publisher = Comment. Soc. regiae sci, Göttingen 7
+  | location = Göttingen
+  | year = 1832}}
+These are in Gauss's ''Werke'', Vol II, pp.&nbsp;65–92 and 93–148; German translations are pp.&nbsp;511–533 and 534–586 of the German edition of the ''Disquisitiones''.
+* {{Citation
+ | last=Baker
+ | first=Alan
+ | author-link=
+ | year=1984
+ | title=A Concise Introduction to the Theory of Numbers
+ | place=Cambridge, UK
+ | publisher=Cambridge University Press
+ | isbn=978-0-521-28654-1
+}}
+* {{Citation
+ | last1=Hardy
+ | first1=G. H.
+ | author1-link=G. H. Hardy
+ | last2=Wright
+ | first2=E. M.
+ | author2-link=E. M. Wright
+ | year=1979
+ | title=An Introduction to the Theory of Numbers
+ | edition=fifth
+ | place=USA
+ | publisher=Oxford University Press
+ | isbn=978-0-19-853171-5
+}}
+* {{Citation
+ | author1 = A. Kornilowicz
+ | author2 = P. Rudnicki
+ | year = 2004
+ | title = Fundamental theorem of arithmetic
+ | journal = [[Formalized Mathematics]]
+ | volume = 12
+ | issue = 2
+ | pages = 179–185
+}}
+* {{citation
+ | first1 = Calvin T.
+ | last1 = Long
+ | year = 1972
+ | title = Elementary Introduction to Number Theory
+ | edition = 2nd | publisher = [[D. C. Heath and Company]]
+ | location = Lexington
+ | lccn = 77-171950
+}}.
+* {{citation
+ | first1 = Anthony J.
+ | last1 = Pettofrezzo
+ | first2 = Donald R.
+ | last2 = Byrkit
+ | year = 1970
+ | title = Elements of Number Theory
+ | publisher = [[Prentice Hall]]
+ | location = Englewood Cliffs
+ | lccn = 77-81766
+}}.
+* {{citation
+  | last1 = Riesel  | first1 = Hans
+  | title = Prime Numbers and Computer Methods for Factorization (second edition)
+  | publisher = Birkhäuser
+  | location = Boston
+  | year = 1994
+   | isbn = 0-8176-3743-5}}
+* {{mathworld|urlname=AbnormalNumber|title=Abnormal number}}
+* {{mathworld|urlname=FundamentalTheoremofArithmetic|title=Fundamental Theorem of Arithmetic}}
+== External links ==
+* [http://www.cut-the-knot.org/blue/gcd_fta.shtml GCD and the Fundamental Theorem of Arithmetic] at [[cut-the-knot]].
+* [http://planetmath.org/fundamentaltheoremofarithmeticproofofthe PlanetMath: Proof of fundamental theorem of arithmetic]
+* [http://fermatslasttheorem.blogspot.com/2005/06/unique-factorization.html Fermat's Last Theorem Blog: Unique Factorization], a blog that covers the history of [[Fermat's Last Theorem]] from [[Diophantus of Alexandria]] to the proof by [[Andrew Wiles]].
+* [http://demonstrations.wolfram.com/FundamentalTheoremOfArithmetic/ "Fundamental Theorem of Arithmetic"] by Hector Zenil, [[Wolfram Demonstrations Project]], 2007.
+* {{cite web|last=Grime|first=James|title=1 and Prime Numbers|url=http://www.numberphile.com/videos/1notprime.html|work=Numberphile|publisher=[[Brady Haran]]}}
+{{Fundamental theorems}}
+{{Divisor classes}}
+[[Category:Theorems about prime numbers]]
+[[Category:Articles containing proofs]]
+[[Category:Fundamental theorems|Arithmetic]]
+[[de:Primfaktorzerlegung#Fundamentalsatz der Arithmetik]]

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Revision as of 17:47, 2 February 2014

History

Applications

Canonical representation of a positive integer

Arithmetic operations

Arithmetical functions

Proof

Existence

Uniqueness

Elementary proof of uniqueness

Generalizations

See also

Notes

References

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