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| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>When you liked this short article and you desire to get more details with regards to [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] i implore you to go to our web site. |
| {{Infobox integer sequence
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| | named_after = [[Arthur Wieferich]]
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| | publication_year = 1909
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| | author = [[Arthur Wieferich|Wieferich, A.]]
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| | terms_number = 2
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| | parentsequence = {{nowrap|Crandall numbers<ref>{{Citation | last1 = Franco | first1 = Z. | last2 = Pomerance | first2 = C. | author2-link = Carl Pomerance | title = On a conjecture of Crandall concerning the qx+1 problem | journal = Math. Of Comput. | volume = 64 | issue = 211 | pages = 1333–1336 | publisher = American Mathematical Society | year = 1995 | url = http://www.math.dartmouth.edu/~carlp/PDF/paper101.pdf | doi = 10.2307/2153499 | postscript = .}}</ref>}}<br>{{nowrap|[[Wieferich prime#Wieferich numbers|Wieferich numbers]]<ref name="Banks, Luca">{{Citation | last1 = Banks | first1 = W. D. | last2 = Luca | first2 = F. | last3 = Shparlinski | first3 = I. E. | title = Estimates for Wieferich numbers | journal = The Ramanujan Journal | volume = 14 | issue = 3 | pages = 361–378
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| | publisher = Springer | year = 2007 | url = http://web.science.mq.edu.au/~igor/Wieferich.pdf
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| | doi = 10.1007/s11139-007-9030-z | postscript = .}}</ref>}}<br>{{nowrap|[[Wieferich prime#Lucas-Wieferich primes|Lucas-Wieferich primes]]<ref name="McIntosh, 2007">{{Citation | last1 = McIntosh | first1 = R. J. | last2 = Roettger | first2 = E. L. | title = A search for Fibonacci-Wieferich and Wolstenholme primes | journal = Mathematics of Computation | volume = 76 | issue = 260 | pages = 2087–2094 | publisher = [[American Mathematical Society|AMS]] | year = 2007 | url = http://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01955-2/S0025-5718-07-01955-2.pdf | archiveurl = http://www.webcitation.org/5usCeulfL | archivedate = 2010-12-10 | doi = 10.1090/S0025-5718-07-01955-2}}</ref>}}<br>{{nowrap|[[Wieferich prime#Near-Wieferich primes|near-Wieferich primes]]}}
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| | first_terms = [[1093 (number)|1093]], [[3511 (number)|3511]]
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| | largest_known_term = [[3511 (number)|3511]]
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| | OEIS = A001220
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| }}
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| In [[number theory]], a '''Wieferich prime''' is a [[prime number]] ''p'' such that ''p''<sup>2</sup> divides 2<sup>''p'' − 1</sup> − 1,<ref name="The Prime Glossary">{{Citation |url=http://primes.utm.edu/glossary/xpage/WieferichPrime.html |title=The Prime Glossary: Wieferich prime }}</ref> therefore connecting these primes with [[Fermat's little theorem]], which states that every odd prime ''p'' divides 2<sup>''p'' − 1</sup> − 1. Wieferich primes were first described by [[Arthur Wieferich]] in 1909 in works pertaining to [[Fermat's last theorem]], at which time both of Fermat's theorems were already well known to mathematicians.<ref name=Kleiner>{{Citation|author=Israel Kleiner |title=From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem |journal=Elem. Math. |volume=55 |year=2000 |page=21 |url=http://math.stanford.edu/~lekheng/flt/kleiner.pdf | doi = 10.1007/PL00000079|postscript=.}}{{WebCite|url=http://www.webcitation.org/5rBbbEvIz}}</ref><ref name=Euler>{{Citation|author=Leonhard Euler |title=Theorematum quorundam ad numeros primos spectantium demonstratio |journal=Novi Comm. Acad. Sci. Petropol. |volume=8 |year=1736 |language=Latin |pages=33–37 |url=http://www.math.dartmouth.edu/~euler/docs/translations/E054tr.pdf|postscript=.}}</ref>
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| Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as [[Mersenne number|Mersenne]] and [[Fermat number|Fermat]] numbers, specific types of [[pseudoprime]]s and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as [[number field]]s and the [[abc conjecture]].
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| {{Asof|2013|October}}, the only known Wieferich primes are 1093 and 3511 {{OEIS|id=A001220}}.
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| == Equivalent definitions ==
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| The stronger version of [[Fermat's little theorem]], which a Wieferich prime satisfies, is usually expressed as a [[congruence relation]] {{nowrap|2<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}}. From the definition of the [[Modular arithmetic#Congruence relation|congruence relation on integers]], it follows that this property is equivalent to the definition given at the beginning. Thus if a prime ''p'' satisfies this congruence, this prime divides the [[Fermat quotient]] <math>\tfrac{2^{p-1}-1}{p}</math>. The following are two illustrative examples using the primes 11 and 1093:
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| : For ''p'' = 11, we get <math>\tfrac{2^{10}-1}{11}</math> which is 93 and leaves a [[remainder]] of 5 after division by 11, hence 11 is not a Wieferich prime. For ''p'' = 1093, we get <math>\tfrac{2^{1092}-1}{1093}</math> or 485439490....2893958515 (307 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime.
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| Wieferich primes can be defined by other equivalent congruences. If ''p'' is a Wieferich prime, one can multiply both sides of the congruence 2<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>2</sup>) by 2 to get 2<sup>''p''</sup> ≡ 2 (mod ''p''<sup>2</sup>). Raising both sides of the congruence to the power ''p'' shows that a Wieferich prime also satisfies 2<sup>''p''<sup>2</sup></sup> ≡2<sup>''p''</sup> ≡ 2 (mod ''p''<sup>2</sup>), and hence 2<sup>''p''<sup>k</sup></sup> ≡ 2 (mod ''p''<sup>2</sup>) for all ''k'' ≥ 1. The converse is also true: 2<sup>''p''<sup>k</sup></sup> ≡ 2 (mod ''p''<sup>2</sup>) for some ''k'' ≥ 1 implies that the [[multiplicative order]] of 2 modulo ''p''<sup>2</sup> divides [[Greatest common divisor|gcd]](''p''<sup>k</sup>-1,[[Euler's totient function|φ]](''p''<sup>2</sup>))=''p''-1, that is, 2<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>2</sup>) and thus ''p'' is a Wieferich prime. This also implies that Wieferich primes can be defined as primes ''p'' such that the multiplicative orders of 2 modulo ''p'' and modulo ''p''<sup>2</sup> coincide: {{nowrap|1=ord<sub>''p''<sup>2</sup></sub> 2 = ord<sub>''p''</sub> 2}}.
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| [[Harry Vandiver|H. S. Vandiver]] proved that {{nowrap|1=2<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>3</sup>)}} if and only if <math>1 + \tfrac{1}{3} + \dots + \tfrac{1}{p-2} \equiv 0 \pmod{p^2}</math>.<ref>{{citation | first=L. E. | last=Dickson | title=Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers | journal=Annals of Mathematics | volume=18 | issue=4 | year=1917 | pages=161–187| url=http://www.jstor.org/stable/pdfplus/2007234}}</ref>{{rp|187}}
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| == History and search status ==
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| In 1902, W. F. Meyer proved a theorem about solutions of the congruence ''a''<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>r</sup>).<ref name=Solutions>{{Citation |author=Wilfrid Keller; Jörg Richstein |title=Solutions of the congruence ''a''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>''r''</sup>) |journal=Math. Comp. |volume=74 |year=2005 |pages=927–936 |doi=10.1090/S0025-5718-04-01666-7 |url=http://www.ams.org/journals/mcom/2005-74-250/S0025-5718-04-01666-7/S0025-5718-04-01666-7.pdf |postscript=. |issue=250}}</ref>{{rp|930}} Later in that decade [[Arthur Wieferich]] showed specifically that if the [[first case of Fermat's last theorem]] has solutions for an odd prime exponent, then that prime must satisfy that congruence for ''a'' = 2 and ''r'' = 2. In other words, if there exist solutions to ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> + ''z''<sup>''p''</sup> = 0 in integers ''x'', ''y'', ''z'' and ''p'' an [[odd prime]] with ''p'' [[List of mathematical symbols#notdivide|∤]] ''xyz'', then ''p'' satisfies 2<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>). In 1913, [[Paul Gustav Heinrich Bachmann|Bachmann]] examined the [[Remainder|residues]] of <math>\tfrac{2^{p-1}-1}{p}\,\bmod\,p</math>. He asked the question when this residue [[Zero of a function|vanishes]] and tried to find expressions for answering this question.<ref>{{cite journal | last = Bachmann | first = P. | title = Über den Rest von <math>\tfrac{2^{p-1}-1}{p}\,\bmod\,p</math> | journal = Journal für Mathematik | volume = 142 | issue = 1 | pages = 41–50 | language=German | year = 1913 | url = http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167646}}</ref>
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| The prime 1093 was found to be a Wieferich prime by Waldemar Meissner in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of <math>\tfrac{2^{t}-1}{p}\,\bmod\,p</math> for all primes ''p'' < 2000 and found this residue to be zero for ''t'' = 364 and ''p'' = 1093, thereby providing a counterexample to a conjecture by Grawe about the impossibility of the Wieferich congruence.<ref name=Meissner>{{citation | first=W. |last=Meissner |title=Über die Teilbarkeit von 2<sup>''p''</sup> − 2 durch das Quadrat der Primzahl ''p''=1093 | journal=Sitzungsber. D. Königl. Preuss. Akad. D. Wiss. | volume=Zweiter Halbband. Juli bis Dezember | place=Berlin | language=German | year=1913 | pages=663–667 }}</ref> E. Haentzschel later ordered verification of the correctness of Meissners congruence via only elementary calculations.<ref>{{citation | first=E. | last=Haentzschel | title=Über die Kongruenz 2<sup>1092</sup> ≡ 1 (mod 1093<sup>2</sup>) | journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]] | volume=34 | year=1926 | language=German | page=284 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00212534X}}</ref>{{rp|664}} Inspired by an earlier work of [[Leonhard Euler|Euler]], he simplified Meissners proof by showing that 1093<sup>2</sup> | (2<sup>182</sup> + 1) and remarked that (2<sup>182</sup> + 1) is a factor of (2<sup>364</sup> − 1).<ref>{{citation | first=E. | last=Haentzschel | title=Über die Kongruenz 2<sup>1092</sup> ≡ 1 (mod 1093<sup>2</sup>) | journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]] | volume=34 | year=1925 | language=German | page=184 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002127695}}</ref> It was also shown that it is possible to prove that 1093 is a Wieferich prime without using [[complex number]]s contrary to the method used by Meissner,<ref>{{citation | first=P. | last=Ribenboim | authorlink=Paulo Ribenboim | title=1093 | journal=The Mathematical Intelligencer | volume=5 | issue=2 | year=1983 | pages=28–34 | doi=10.1007/BF03023623}}</ref> although Meissner himself hinted at that he was aware of a proof without complex values.<ref name=Meissner />{{rp|665}}
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| The prime [[3511 (number)|3511]] was first found to be a Wieferich prime by [[N. G. W. H. Beeger]] in 1922<ref>{{citation | first=N. G. W. H. |last=Beeger | authorlink=N. G. W. H. Beeger | title=On a new case of the congruence 2<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>) | journal=[[Messenger of Mathematics]] | volume=51 | year=1922 | pages=149–150 |url=http://www.archive.org/stream/messengerofmathe5051cambuoft#page/148/mode/2up}}{{WebCite|url=http://www.webcitation.org/5zo7d2jde}}</ref> and another proof of it being a Wieferich prime was published in 1965 by [[Richard K. Guy|Guy]].<ref>{{citation | first=R. K. | last=Guy | authorlink=Richard K. Guy | title=A property of the prime 3511 | journal=The Mathematical Gazette | volume=49 | issue=367 | year=1965 | pages=78–79 | url=http://www.jstor.org/stable/3614249}}</ref> In 1960, Kravitz<ref>{{cite journal |author=Kravitz, S. |url=http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0121334-7/S0025-5718-1960-0121334-7.pdf |title=The Congruence 2<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>2</sup>) for ''p'' < 100,000 |journal=Math. Comp. |volume=14 |year=1960 |pages=378 |doi=10.1090/S0025-5718-1960-0121334-7}}</ref> doubled a previous record set by Fröberg<ref>{{cite journal |author=Fröberg C. E. |url=http://www.ams.org/journals/mcom/1958-12-064/S0025-5718-58-99270-6/S0025-5718-58-99270-6.pdf |title=Some Computations of Wilson and Fermat Remainders |journal=Math. Comp. |volume=12 |year=1958 |pages=281 |doi=10.1090/S0025-5718-58-99270-6 }}</ref> and in 1961 Riesel extended the search to 500000 with the aid of [[BESK]].<ref>{{cite journal |author=[[Hans Riesel|Riesel, H.]] |url=http://www.ams.org/journals/mcom/1964-18-085/S0025-5718-1964-0157928-6/S0025-5718-1964-0157928-6.pdf |title=Note on the Congruence ''a''<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>2</sup>) |journal=Math. Comp. |volume=18 |year=1964 |pages=149–150 |doi=10.1090/S0025-5718-1964-0157928-6}}</ref> Around 1980, Lehmer was able to reach the search limit of 6{{e|9}}.<ref>{{cite journal | last = Lehmer | first = D. H. | authorlink = Derrick Henry Lehmer | title = On Fermat's quotient, base two | journal = Math. Comp. | volume = 36 | issue = 153 | pages = 289–290 | url = http://www.ams.org/journals/mcom/1981-36-153/S0025-5718-1981-0595064-5/S0025-5718-1981-0595064-5.pdf |doi=10.1090/S0025-5718-1981-0595064-5}}</ref> This limit was extended to over 2.5{{e|15}} in 2006,<ref name="Ribenboim, 2004" /> finally reaching 3{{e|15}}. It is now known, that if any other Wieferich primes exist, they must be greater than 6.7{{e|15}}.<ref name=Dorais>{{cite journal | last = Dorais | first = F. G. | coauthors = Klyve, D. | title = A Wieferich Prime Search Up to 6.7{{e|15}} | journal = Journal of Integer Sequences | volume = 14 | issue = 9 | year = 2011 | url = http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.pdf | zbl = 05977305 | accessdate = 2011-10-23}}</ref> The search for new Wieferich primes is currently performed by the [[distributed computing]] project Wieferich@Home. In December 2011, another search was started by the [[PrimeGrid]] project.<ref>PrimeGrid [http://www.primegrid.com/forum_thread.php?id=3894 Announcement of Wieferich and Wall-Sun-Sun searches]</ref> {{As of|2013|October}}, PrimeGrid has extended the search limit to 1.2{{e|17}} and continues.<ref>PrimeGrid [http://prpnet.mine.nu:13000/ Wieferich prime search server statistics]</ref>
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| It has been conjectured (as for [[Wilson prime]]s) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below ''x'' is approximately log(log(''x'')), which is a [[heuristic argument|heuristic result]] that follows from the plausible assumption that for a prime ''p'', the (''p'' − 1)-th degree [[roots of unity]] modulo ''p''<sup>2</sup> are [[uniform distribution (discrete)|uniformly distributed]] in the [[multiplicative group of integers modulo n|multiplicative group of integers modulo ''p''<sup>2</sup>]].<ref name=Crandall />
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| == Properties ==
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| === Connection with Fermat's last theorem ===
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| The following theorem connecting Wieferich primes and [[Fermat's last theorem]] was proven by Wieferich in 1909:<ref>{{Citation | first=A. |last=Wieferich | authorlink=Arthur Wieferich | title=Zum letzten Fermat'schen Theorem | journal=[[Journal für die reine und angewandte Mathematik]] | volume=136 | year=1909 | language=German | pages=293–302 |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002166968 | doi=10.1515/crll.1909.136.293 | issue=136 | postscript=.}}</ref>
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| :Let ''p'' be prime, and let ''x'', ''y'', ''z'' be [[integer]]s such that ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> + ''z''<sup>''p''</sup> = 0. Furthermore, assume that ''p'' does not divide the [[product (mathematics)|product]] ''xyz''. Then ''p'' is a Wieferich prime.
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| The above case (where ''p'' does not divide any of ''x'', ''y'' or ''z'') is commonly known as the [[first case of Fermat's last theorem]] (FLTI)<ref>{{Citation | last=Coppersmith | first=D. | authorlink=Don Coppersmith | title=Fermat's Last Theorem (Case I) and the Wieferich Criterion | journal=Math. Comp. | volume=54 | issue=190 | pages=895–902 | publisher=[[American Mathematical Society|AMS]] | year=1990 | jstor=2008518 |url=http://www.ams.org/journals/mcom/1990-54-190/S0025-5718-1990-1010598-2/S0025-5718-1990-1010598-2.pdf | postscript=. }}</ref><ref>{{Citation | last=Cikánek | first=P. | title=A Special Extension of Wieferich's Criterion | journal=Math. Comp. | volume=62 | issue=206 | pages=923–930 | jstor=3562296 | publisher=[[American Mathematical Society|AMS]] | year=1994 | url=http://www.ams.org/journals/mcom/1994-62-206/S0025-5718-1994-1216257-9/S0025-5718-1994-1216257-9.pdf | postscript=.}}</ref> and FLTI is said to fail for a prime ''p'', if solutions to the Fermat equation exist for that ''p'', otherwise FLTI holds for ''p''.<ref name=Dilcher,Skula>{{Citation | last1=Dilcher | first1=K. | last2=Skula | first2=L. | title=A new criterion for the first case of Fermat's last theorem | journal=Math. Comp. | volume=64 | issue=209 | pages=363–392 | jstor=2153341 | publisher=AMS | year=1995 | url=http://www.ams.org/journals/mcom/1995-64-209/S0025-5718-1995-1248969-6/S0025-5718-1995-1248969-6.pdf}}</ref>
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| In 1910, [[Mirimanoff]] expanded<ref>{{Citation | first=D. |last=Mirimanoff |title=Sur le dernier théorème de Fermat | language=French | journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences | volume=150 | year=1910 | pages=293–206 | postscript=. }}</ref> the theorem by showing that, if the preconditions of the theorem hold true for some prime ''p'', then ''p''<sup>2</sup> must also divide 3<sup>''p'' − 1</sup> − 1. Granville and Monagan further proved that ''p''<sup>2</sup> must actually divide ''m''<sup>''p'' − 1</sup> − 1 for every prime ''m'' ≤ 89.<ref name="Granville, Monagan">{{Citation | first1=A. | last1=Granville | first2=M. B. | last2=Monagan | title=The First Case of Fermat's Last Theorem is true for all prime exponents up to 714,591,416,091,389 | journal=Transactions of the American Mathematical Society | volume=306 | issue=1 | year=1988 | pages=329–359 | doi=10.1090/S0002-9947-1988-0927694-5 | postscript=.}}</ref> Suzuki extended the proof to all primes ''m'' ≤ 113.<ref>{{citation | first=Jiro |last=Suzuki | title=On the generalized Wieferich criteria | journal=Proc. Japan Acad. Ser. A Math. Sci. | volume=70 | pages=230–234 | year=1994 | url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195510946 }}</ref>
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| Let ''H<sub>p</sub>'' be a set of pairs of integers with 1 as their [[greatest common divisor]], ''p'' being prime to ''x'', ''y'' and ''x'' + ''y'', (''x'' + ''y'')<sup>''p''-1</sup> ≡ 1 (mod p<sup>2</sup>), (''x'' + ''ξy'') being the ''p''th power of an [[ideal (ring theory)|ideal]] of ''K'' with ''ξ'' defined as cos 2''π''/''p'' + ''i'' sin 2''π''/''p''. ''K'' = '''Q'''(''ξ'') is the [[field extension]] obtained by adjoining all [[polynomial]]s in the [[algebraic number]] ''ξ'' to the [[Field (mathematics)|field]] of [[rational number]]s (such an extension is known as a [[Algebraic number field|number field]] or in this particular case, where ''ξ'' is a [[root of unity]], a [[cyclotomic field|cyclotomic number field]]).<ref name="Granville, Monagan" />{{rp|332}}
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| From [[Fundamental theorem of ideal theory in number fields|uniqueness of factorization of ideals in '''Q'''(ξ)]] it follows that if the first case of Fermat's last theorem has solutions ''x'', ''y'', ''z'' then ''p'' divides ''x''+''y''+''z'' and (''x'', ''y''), (''y'', ''z'') and (''z'', ''x'') are elements of ''H<sub>p</sub>''.<ref name="Granville, Monagan" />{{rp|333}}
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| Granville and Monagan showed that (1, 1) ∈ ''H<sub>p</sub>'' if and only if ''p'' is a Wieferich prime.<ref name="Granville, Monagan" />{{rp|333}}
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| === Connection with the ''abc'' conjecture and non-Wieferich primes ===
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| A non-Wieferich prime is a prime ''p'' satisfying 2<sup>''p'' − 1</sup> ≢ 1 (mod ''p''<sup>2</sup>). [[Joseph H. Silverman|J. H. Silverman]] showed in 1988 that if the [[abc conjecture]] holds, then there exist infinitely many non-Wieferich primes.<ref>Charles, D. X. [http://pages.cs.wisc.edu/~cdx/Criterion.pdf On Wieferich primes]</ref> More precisely he showed that the abc conjecture implies the existence of a constant only depending on ''α'' such that the number of non-Wieferich primes to base ''α'' with ''p'' less than or equal to a variable ''X'' is greater than log(''X'') as ''X'' goes to infinity.<ref>{{citation | first=J. H. |last=Silverman | authorlink=Joseph H. Silverman | title=Wieferich's criterion and the abc-conjecture | journal=[[Journal of Number Theory]] | volume=30 | issue=2 | year=1988 | pages=226–237 | doi=10.1016/0022-314X(88)90019-4 }}</ref>{{rp|227}} Numerical evidence suggests that very few of the prime numbers in a given interval are Wieferich primes. The set of Wieferich primes and the set of non-Wieferich primes, sometimes denoted by ''W<sub>2</sub>'' and ''W<sub>2</sub><sup>c</sup>'' respectively,<ref name=DeKoninckDoyon>{{citation | first1=J.-M. | last1=DeKoninck | first2=N. | last2=Doyon | title=On the set of Wieferich primes and of its complement | journal=Annales Univ. Sci. Budapest., Sect. Comp. | volume=27 | year=2007 | pages=3–13 | url=http://ac.inf.elte.hu/Vol_027_2007/003.pdf}}</ref> are [[Complement (set theory)|complementary sets]], so if one of them is shown to be finite, the other one would necessarily have to be infinite, because both are [[Subset#Definitions|proper subsets]] of the set of prime numbers. It was later shown that the existence of infinitely many non-Wieferich primes already follows from a weaker version of the abc conjecture, called the ''ABC-(k, ε) conjecture''.<ref>{{citation | first=K. | last=Broughan | title=Relaxations of the ABC Conjecture using integer ''k'' 'th roots | journal=New Zealand J. Math. | volume=35 | issue=2 | year=2006 | pages=121–136 | url=http://www.math.waikato.ac.nz/~kab/papers/abc01.pdf}}</ref> Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers<ref>{{cite book | last = Ribenboim | first = P. | authorlink = Paulo Ribenboim | title = 13 Lectures on Fermat's Last Theorem
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| | publisher = Springer | year = 1979 | location = New York | page = 154 | url = http://books.google.de/books?id=w2o67-lBGhIC&lpg=PP1&dq=13%20lectures%20on%20Fermat’s%20last%20theorem&hl=de&pg=PA154#v=onepage&q=%22Let%20me%20add%20that%20Schinzel%20has%20conjectured%22&f=false | isbn = 0-387-90432-8}}</ref> as well as if there exists a real number ''ξ'' such that the set {''n'' ∈ '''N''' : λ(2<sup>''n''</sup> − 1) < 2 − ''ξ''} is of [[Natural density|density]] one, where the ''index of composition'' ''λ''(''n'') of an integer ''n'' is defined as <math>\tfrac{\log n}{\log \gamma (n)}</math> and <math style="vertical-align:-65%;">\gamma (n) = \prod_{p \mid n} p</math>, meaning <math>\gamma (n)</math> gives the product of all [[prime factor]]s of ''n''.<ref name=DeKoninckDoyon />{{rp|4}}
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| === Connection with Mersenne and Fermat primes ===
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| It is known that the ''n''th [[Mersenne number]] ''M''<sub>''n''</sub> = 2<sup>''n''</sup> − 1 is prime only if ''n'' is prime. [[Fermat's little theorem]] implies that if ''p'' > 2 is prime, then ''M''<sub>''p''−1</sub> (= 2<sup>''p'' − 1</sup> − 1) is always divisible by ''p''. Since Mersenne numbers of prime indices ''M''<sub>''p''</sub> and ''M''<sub>''q''</sub> are co-prime,
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| ::A prime divisor ''p'' of ''M''<sub>''q''</sub>, where ''q'' is prime, is a Wieferich prime if and only if ''p''<sup>2</sup> divides ''M''<sub>''q''</sub>.<ref>{{Citation |url=http://primes.utm.edu/mersenne/index.html#unknown |title=Mersenne Primes: Conjectures and Unsolved Problems }}</ref>
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| Thus, a Mersenne prime cannot also be a Wieferich prime. A notable [[Unsolved problems in mathematics|open problem]] is to determine whether or not all Mersenne numbers of prime index are [[Square-free integer|square-free]]. If ''q'' is prime and the Mersenne number ''M''<sub>''q''</sub> is ''not'' square-free, that is, there exists a prime ''p'' for which ''p''<sup>2</sup> divides ''M''<sub>''q''</sub>, then ''p'' is a Wieferich prime. Therefore, if there are only finitely many Wieferich primes, then there will be at most finitely many Mersenne numbers with prime index that are not square-free. Rotkiewicz showed a related result: if there are infinitely many square-free Mersenne numbers, then there are infinitely many non-Wieferich primes.<ref>{{cite journal | last = Rotkiewicz | first = A. | title = Sur les nombres de Mersenne dépourvus de diviseurs carrés et sur les nombres naturels n, tels que n<sup>2</sup>∣2<sup>n</sup>-2 | journal = Mat. Vesnik | volume = 2 | issue = 17 | pages = 78–80 | year = 1965 | language = French | url = http://resolver.sub.uni-goettingen.de/purl?PPN311571026_0017}}</ref>
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| Similarly, if ''p'' is prime and ''p''<sup>2</sup> divides some [[Fermat number]] ''F''<sub>''n''</sub> = 2<sup>2<sup>''n''</sup></sup> + 1, then ''p'' must be a Wieferich prime.<ref>{{Citation |authorlink=Paulo Ribenboim |last=Ribenboim |first=Paulo |title=The little book of big primes |place=New York |publisher=Springer |year=1991 |page=64 |isbn=0-387-97508-X |url=http://books.google.com/?id=zUCK7FT4xgAC&pg=PA64 }}</ref>
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| For the primes 1093 and 3511, it was shown that neither of them is a divisor of any Mersenne number with prime index nor a divisor of any Fermat number.<ref>{{citation | first1=H. G. | last1=Bray | first2=L. J. | last2=Warren | title=On the square-freeness of Fermat and Mersenne numbers | journal=Pacific J. Math. | volume=22 | issue=3 | year=1967 | pages=563–564 | url=http://projecteuclid.org/euclid.pjm/1102992105 | mr=0220666 | zbl=0149.28204}}</ref>
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| === Connection with other equations ===
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| Scott and Styer showed that the equation ''p''<sup>x</sup> – 2<sup>y</sup> = ''d'' has at most one solution in positive integers (''x'', ''y''), unless when ''p''<sup>4</sup> | 2<sup>ord<sub>''p''</sub> 2</sup> – 1 if ''p'' ≢ 65 (mod 192) or unconditionally when ''p''<sup>2</sup> | 2<sup>ord<sub>''p''</sub> 2</sup> – 1, where ord<sub>''p''</sub> 2 denotes the [[multiplicative order]] of 2 modulo ''p''.<ref>{{cite journal | last = Scott | first = R. | coauthors = Styer, R. | title = On p<sup>x</sup>-q<sup>y</sup>=c and related three term exponential Diophantine equations with prime bases | journal = Journal of Number Theory | volume = 105 | issue = 2 | pages = 212–234 | publisher = Elsevier | date = April 2004 | doi = 10.1016/j.jnt.2003.11.008}}</ref>{{rp|215, 217–218}} They also showed that a solution to the equation ±a<sup>x<sub>1</sub></sup> ± 2<sup>y<sub>1</sub></sup> = ±a<sup>x<sub>2</sub></sup> ± 2<sup>y<sub>2</sub></sup> = ''c'' must be from a specific set of equations but that this does not hold, if ''a'' is a Wieferich prime greater than 1.25 x 10<sup>15</sup>.<ref>{{cite journal | last = Scott | first = R. | coauthors = Styer, R. | title = On the generalized Pillai equation ±''a''<sup>''x''</sup>±''b''<sup>''y''</sup>=''c'' | journal = Journal of Number Theory | volume = 118 | issue = 2 | pages = 236–265 | year = 2006 | url = http://www.sciencedirect.com/science/article/pii/S0022314X05002064/pdf?md5=dee9204a33938b3cea0600f3480ca133&pid=1-s2.0-S0022314X05002064-main.pdf | doi = 10.1016/j.jnt.2005.09.001}}</ref>{{rp|258}}
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| === Binary periodicity of ''p''−1 ===
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| Johnson observed<ref name=John>{{Citation |author=Wells Johnson |title=On the nonvanishing of Fermat quotients (mod ''p'') |journal=J. Reine angew. Math. |volume=292 |year=1977 |pages=196–200 |url=http://www.digizeitschriften.de/index.php?id=resolveppn&PPN=GDZPPN002193698}}</ref> that the two known Wieferich primes are one greater than numbers with periodic [[binary expansion]]s (1092 = 010001000100<sub>2</sub>; 3510 = 110110110110<sub>2</sub>). The Wieferich@Home project searches for Wieferich primes by testing numbers that are one greater than a number with a periodic binary expansion, but up to a "bit pseudo-length" of 3500 of the tested binary numbers generated by combination of bit strings with a bit length of up to 24 it has not found a new Wieferich prime.<ref name=DoKu>{{Citation |last1=Dobeš |first1=Jan |last2=Kureš |first2=Miroslav |title=Search for Wieferich primes through the use of periodic binary strings |journal=Serdica Journal of Computing |volume=4 |year=2010 |pages=293–300 |url=http://sci-gems.math.bas.bg/jspui/bitstream/10525/1595/1/sjc104-vol4-num3-2010.pdf |zbl=05896729 |postscript=.}}</ref>
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| === Connection with pseudoprimes ===
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| It was observed that the two known Wieferich primes are the square factors of all [[Square-free integer|non-square free]] base-2 [[Fermat pseudoprime]]s up to 25{{e|9}}.<ref>{{Citation | author1-link=Paulo Ribenboim | last1=Ribenboim | first1=P. | title=The Little Book of Bigger Primes | location=New York | publisher=Springer-Verlag New York, Inc. | year=2004 | isbn=0-387-20169-6 | page=99 |chapter=[http://trex58.files.wordpress.com/2010/01/ribenboimbook1.pdf Chapter 2. How to Recognize Whether a Natural Number is a Prime]}} {{WebCite|url=http://www.webcitation.org/5uTrtgMk3}}</ref> Later computations showed that the only repeated factors of the pseudoprimes up to 10<sup>12</sup> are 1093 and 3511.<ref>{{cite journal | last = Pinch | first = R. G. E. | title = The Pseudoprimes up to 10<sup>13</sup> | journal = Lecture Notes in Computer Science | volume = 1838 | pages = 459–473 | year = 2000 | url = http://130.203.133.150/viewdoc/download?doi=10.1.1.32.3849&rep=rep1&type=pdf | doi = 10.1007/10722028_30}}</ref> In addition, the following connection exists: Let ''n'' be a base 2 pseudoprime and ''p'' be a prime divisor of ''n''. If <math>\tfrac{2^{n-1}-1}{n}\not\equiv 0 \pmod{p}</math>, then also <math>\tfrac{2^{p-1}-1}{p}\not\equiv 0 \pmod{p}</math>.<ref name=Dilcher,Skula />{{rp|378}} Furthermore if ''p'' is a Wieferich prime, then ''p''<sup>2</sup> is a [[Catalan pseudoprime]].<ref>{{cite journal | last1 = Aebi | first1 = C. | last2 = Cairns |first2 = G. | title = Catalan numbers, primes and twin primes | journal = Elemente der Mathematik |volume=63 |issue=4 | pages = 153–164 | year = 2008 | url = http://www.latrobe.edu.au/mathstats/staff/cairns/papers/catalan.pdf | doi = 10.4171/EM/103}} {{WebCite|url=http://www.webcitation.org/6AZIW3nPx}}</ref>
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| === Connection with directed graphs ===
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| For all primes up to 100000 ''L''(''p''<sup>''n''+1</sup>) = ''L''(''p''<sup>''n''</sup>) only for two cases: ''L''(''1093''<sup>''2''</sup>) = ''L''(''1093'') = 364 and ''L''(''3511''<sup>''2''</sup>) = ''L''(''3511'') = 1755, where ''m'' is the modulus of the doubling diagram and ''L''(''m'') gives the number of vertices in the cycle of 1. The term doubling diagram refers to the [[directed graph]] with 0 and the natural numbers less than ''m'' as vertices with arrows pointing from each vertex ''x'' to vertex 2''x'' reduced modulo ''m''.<ref name="Ehrlich">{{Citation | first=A. | last=Ehrlich | title=Cycles in Doubling Diagrams mod m | journal=The Fibonacci Quarterly | volume=32 | issue=1 | year=1994 | pages=74–78 | url=http://www.fq.math.ca/Scanned/32-1/ehrlich.pdf | postscript=.}}</ref>{{rp|74}} It was shown, that for all odd prime numbers either ''L''(''p''<sup>''n''+1</sup>) = ''p'' × ''L''(''p''<sup>''n''</sup>) or ''L''(''p''<sup>''n''+1</sup>) = ''L''(''p''<sup>''n''</sup>).<ref name="Ehrlich" />{{rp|75}}
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| === Properties related to number fields ===
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| It was shown that <math>\chi_{D_{0}} \big(p \big) = 1</math> and <math>\lambda\,\!_p \big( \mathbb{Q} \big(\sqrt{D_{0}} \big) \big) = 1</math> if and only if 2<sup>''p'' − 1</sup> ≢ 1 (mod ''p''<sup>2</sup>) where ''p'' is an odd prime and <math>D_{0} < 0</math> is the [[fundamental discriminant]] of the imaginary [[quadratic field]] <math>\mathbb{Q} \big(\sqrt{1 - p^2} \big)</math>. Furthermore the following was shown: Let ''p'' be a Wieferich prime. If {{nowrap|''p'' ≡ 3 (mod 4)}}, let <math>D_{0} < 0</math> be the fundamental discriminant of the imaginary quadratic field <math>\mathbb{Q} \big(\sqrt{1 - p} \big)</math> and if {{nowrap|''p'' ≡ 1 (mod 4)}}, let <math>D_{0} < 0</math> be the fundamental discriminant of the imaginary quadratic field <math>\mathbb{Q} \big(\sqrt{4 - p} \big)</math>. Then <math>\chi_{D_{0}} \big(p \big) = 1</math> and <math>\lambda\,\!_p \big( \mathbb{Q} \big(\sqrt{D_{0}} \big) \big) = 1</math> (''χ'' and ''λ'' in this context denote Iwasawa [[Invariant (mathematics)|invariants]]).<ref>{{citation | first=D. | last=Byeon | title=Class numbers, Iwasawa invariants and modular forms | journal=Trends in Mathematics | volume=9 | issue=1 | year=2006 | pages=25–29 | url=http://basilo.kaist.ac.kr/mathnet/kms_tex/985999.pdf}}</ref>{{rp|27}}
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| Furthermore the following result was obtained: Let ''q'' be an odd prime number, ''k'' and ''p'' are primes such that {{nowrap|1=''p'' = 2''k'' + 1,}} {{nowrap|''k'' ≡ 3 (mod 4),}} {{nowrap|''p'' ≡ −1 (mod ''q''),}} {{nowrap|''p'' ≢ −1 (mod ''q''<sup>3</sup>)}} and the order of ''q'' modulo ''k'' is <math>\tfrac{k-1}{2}</math>. Assume that ''q'' divides ''h''<sup>+</sup>, the [[Ideal class group|class number]] of the real [[cyclotomic field]] <math>\mathbb{Q} \big( \zeta\,\!_p + \zeta\,\!_p^{-1} \big)</math>, the cyclotomic field obtained by adjoining the sum of a ''p''-th [[root of unity]] and its [[Multiplicative inverse|reciprocal]] to the field of rational numbers. Then ''q'' is a Wieferich prime.<ref name="Jakubec">{{citation | first=S. | last=Jakubec | title=Connection between the Wieferich congruence and divisibility of h<sup>+</sup> | journal=Acta Arithmetica | volume=71 | issue=1 | year=1995 | pages=55–64 | url=http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7114.pdf}}</ref>{{rp|55}} This also holds if the conditions {{nowrap|''p'' ≡ −1 (mod ''q'')}} and {{nowrap|''p'' ≢ −1 (mod ''q''<sup>3</sup>)}} are replaced by {{nowrap|''p'' ≡ −3 (mod ''q'')}} and {{nowrap|''p'' ≢ −3 (mod ''q''<sup>3</sup>)}} as well as when the condition {{nowrap|''p'' ≡ −1 (mod ''q'')}} is replaced by {{nowrap|''p'' ≡ −5 (mod ''q'')}} (in which case ''q'' is a [[Wall−Sun−Sun prime]]) and the incongruence condition replaced by {{nowrap|''p'' ≢ −5 (mod ''q''<sup>3</sup>)}}.<ref>{{citation | first=S. | last=Jakubec | title=On divisibility of the class number h<sup>+</sup> of the real cyclotomic fields of prime degree l | journal=Mathematics of Computation | volume=67 | issue=221 | year=1998 | pages=369–398 | url=http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00916-8/S0025-5718-98-00916-8.pdf}}</ref>{{rp|376}}
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| ==Generalizations==
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| ===Near-Wieferich primes===
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| A prime ''p'' satisfying the congruence 2<sup>(''p''−1)/2</sup> ≡ ±1 + ''Ap'' (mod ''p''<sup>2</sup>) with small |''A''| is commonly called a ''near-Wieferich prime'' {{OEIS|id=A195988}}.<ref name=Crandall>{{Citation | first=Richard E. |last=Crandall |first2=Karl |last2=Dilcher |first3=Carl |last3=Pomerance | title=A search for Wieferich and Wilson primes | journal=Math. Comput. | volume=66 | issue=217 | pages=433–449 | year=1997 | doi=10.1090/S0025-5718-97-00791-6 |url=http://gauss.dartmouth.edu/~carlp/PDF/paper111.pdf | postscript=. }}</ref><ref name=Knauer>{{Citation |author=Joshua Knauer; Jörg Richstein |title=The continuing search for Wieferich primes |journal=Math. Comp. |volume=74 |year=2005 |pages=1559–1563 |doi=10.1090/S0025-5718-05-01723-0 |url=http://www.ams.org/journals/mcom/2005-74-251/S0025-5718-05-01723-0/S0025-5718-05-01723-0.pdf |postscript=. |issue=251}}</ref> Near-Wieferich primes with ''A'' = 0 represent Wieferich primes. Recent searches, in addition to their primary search for Wieferich primes, also tried to find near-Wieferich primes.<ref name=Dorais/><ref name=elmath>[http://www.elmath.org/index.php?id=main About project Wieferich@Home]</ref> The following table lists all near-Wieferich primes with |''A''| ≤ 10 in the interval [1{{e|9}}, 3{{e|15}}].<ref>PrimeGrid, [http://www.primegrid.com/download/wieferich_list.pdf Wieferich & near Wieferich primes p < 11e15]</ref> This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.<ref name="Ribenboim, 2004">{{Citation | last = Ribenboim | first = Paulo | authorlink = Paulo Ribenboim | title = Die Welt der Primzahlen: Geheimnisse und Rekorde | publisher = Springer | year = 2004 | location = New York | page = 237 | language = German | url = http://books.google.de/books?id=-nEM9ZVr4CsC&pg=PA237 | isbn = 3-540-34283-4}}</ref><ref>{{Citation | last = Ribenboim | first = Paulo | authorlink = Paulo Ribenboim | title = My numbers, my friends: popular lectures on number theory | publisher = Springer | year = 2000 | location = New York | pages = 213–229 | isbn = 978-0-387-98911-2}}</ref>
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| {| class="wikitable" style="width:40%; border:0; text-align:right;"
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| |-
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| ! p !! 1 or −1 !! A
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| |-
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| | 3520624567 || +1 || −6
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| |-
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| | 46262476201 || +1 || +5
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| |-
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| | 47004625957 || −1 || +1
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| |-
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| | 58481216789 || −1 || +5
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| |-
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| | 76843523891 || −1 || +1
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| |-
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| | 1180032105761 || +1 || −6
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| |-
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| | 12456646902457 || +1 || +2
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| |-
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| | 134257821895921 || +1 || +10
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| |-
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| | 339258218134349 || −1 || +2
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| |-
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| | 2276306935816523 || −1 || −3
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| |}
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| Dorais and Klyve<ref name=Dorais/> used a different definition of a near-Wieferich prime, defining it as a prime ''p'' with small value of <math>\left|\tfrac{\omega(p)}{p}\right|</math> where <math>\omega(p)=\tfrac{2^{p-1}-1}{p}\,\bmod\,p</math> is the [[Fermat quotient]] of 2 with respect to ''p'' modulo ''p'' (the [[modulo operation]] here gives the residue with the smallest absolute value). The following table lists all primes ''p'' ≤ 6.7 × 10<sup>15</sup> with <math>\left|\tfrac{\omega(p)}p\right|\leq 10^{-14}</math>.
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| {| class="wikitable" style="width:40%; border:0; text-align:right;"
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| |-
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| ! ''p'' !! <math>\omega(p)</math> !! <math>\left|\tfrac{\omega(p)}{p}\right|\times 10^{14}</math>
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| | 1093 || 0 || 0
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| |-
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| | 3511 || 0 || 0
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| |-
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| | 2276306935816523 || +6 || 0.264
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| |-
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| | 3167939147662997 || −17 || 0.537
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| |-
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| | 3723113065138349 || −36 || 0.967
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| |-
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| | 5131427559624857 || −36 || 0.702
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| |-
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| | 5294488110626977 || −31 || 0.586
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| |-
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| | 6517506365514181 || +58 || 0.890
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| |}
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| ===Base-''a'' Wieferich primes===
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| {{Main|Fermat quotient#Generalized Wieferich primes|l1=Fermat quotient}}
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| A ''Wieferich prime base a'' is a prime ''p'' that satisfies
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| : ''a''<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>).<ref name="Solutions"/>
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| Such a prime cannot divide ''a'', since then it would also divide 1.
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| Bolyai showed that if ''p'' and ''q'' are primes, ''a'' is a positive integer not divisible by ''p'' and ''q'' such that {{nowrap|''a''<sup>''p''-1</sup> ≡ 1 (mod ''q'')}}, {{nowrap|''a''<sup>''q''-1</sup> ≡ 1 (mod ''p'')}}, then {{nowrap|''a''<sup>''pq''-1</sup> ≡ 1 (mod ''pq'')}}. Setting ''p'' = ''q'' leads to {{nowrap|''a''<sup>''p''<sup>2</sup>-1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}}.<ref name="Kiss">{{cite journal | last = Kiss | first = E. | coauthors = Sándor, J. | title = On a congruence by János Bolyai, connected with pseudoprimes | journal = Mathematica Pannonica | volume = 15 | issue = 2 | pages = 283–288 | year = 2004 | url = http://ttk.pte.hu/mii/html/pannonica/index_elemei/mp15-2/mp15-2-283-288.pdf}}</ref>{{rp|284}} It was shown that {{nowrap|''a''<sup>''p''<sup>2</sup>-1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}} if and only if {{nowrap|''a''<sup>''p''-1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}}.<ref name="Kiss" />{{rp|285–286}}
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| ===Wieferich pairs===
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| {{main|Wieferich pair}}
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| A [[Wieferich pair]] is a pair of primes ''p'' and ''q'' that satisfy
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| : ''p''<sup>''q'' − 1</sup> ≡ 1 (mod ''q''<sup>2</sup>) and ''q''<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>)
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| so that a Wieferich prime ''p'' ≡ 1 (mod 4) will form such a pair (''p'', 2): the only known instance in this case is ''p'' = 1093. There are 7 known Wieferich pairs.<ref>{{mathworld| |urlname=DoubleWieferichPrimePair |title=Double Wieferich Prime Pair}}</ref>
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| ===Wieferich numbers===
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| A '''Wieferich number''' is an odd integer ''w'' ≥ 3 satisfying the congruence 2<sup>''φ''(''w'')</sup> ≡ 1 (mod ''w''<sup>2</sup>), where ''φ''(·) denotes the [[Euler's totient function|Euler function]]. If Wieferich number ''w'' is prime, then it is a Wieferich prime. The first few Wieferich numbers are:
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| : 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, … {{OEIS|A077816}}
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| It can be shown that if there are only finitely many Wieferich primes, then there are only finitely many Wieferich numbers. In particular, if the only Wieferich primes are 1093 and 3511, then there exist exactly 104 Wieferich numbers, which matches the number of Wieferich numbers currently known.<ref name="Banks, Luca"/>
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| More generally, an integer ''w'' is a '''Wieferich number to base''' ''a'', if ''a''<sup>''φ''(''w'')</sup> ≡ 1 (mod ''w''<sup>2</sup>).<ref>{{citation | first1=T. | last1=Agoh | first2=K. | last2=Dilcher | first3=L. | last3=Skula | title=Fermat Quotients for Composite Moduli | journal=Journal of Number Theory | volume=66 | issue=1 | year=1997 | pages=29–50 | doi=10.1006/jnth.1997.2162}}</ref>{{rp|31}}
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| Another definition specifies a '''Wieferich number''' as positive odd integer ''q'' such that ''q'' and <math>\tfrac{2^m-1}{q}</math> are not [[coprime]], where ''m'' is the [[multiplicative order]] of 2 modulo ''q''. The first of these numbers are:<ref>{{cite journal | last = Müller | first = H. | journal = Mitteilungen der Mathematischen Gesellschaft in Hamburg | volume = 28 | pages = 121–130 | year = 2009 | language = German | publisher = Mathematische Gesellschaft in Hamburg | chapter = Über Periodenlängen und die Vermutungen von Collatz und Crandall | url = http://books.google.com/books?ei=BTWeT5XzF_TV4QTTurGqDg&hl=de&id=hgYTAQAAMAAJ&dq=wieferich+21%2C+39%2C+55%2C+57%2C+105%2C+111%2C+147%2C+155%2C+165%2C+171%2C+183&q=21%2C+39%2C+55%2C+57%2C+105%2C+111%2C+147%2C+155%2C+165%2C+171%2C+183#search_anchor}}</ref>
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| : 21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, … {{OEIS|A182297}}
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| As above, if Wieferich number ''q'' is prime, then it is a Wieferich prime.
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| ===Lucas-Wieferich primes===
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| A '''Lucas-Wieferich prime''' associated with the [[Ordered pair|pair]] of integers ''(P, Q)'' is a prime ''p'' such that ''U<sub>p-ε</sub>(P, Q)'' ≡ 0 (mod ''p''<sup>2</sup>), where ''U<sub>n</sub>(P, Q)'' denotes the [[Lucas sequence]] [[Lucas sequence#Recurrence relations|of the first kind]] and ''ε'' equals the [[Legendre symbol]] of ''P''<sup>2</sup> - 4''Q'' modulo ''p''. All Wieferich primes are Lucas-Wieferich primes associated with the pair (3, 2).<ref name="McIntosh, 2007" />{{rp|2088}}
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| ===Wieferich places===
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| Let ''K'' be a [[global field]], i.e. a [[number field]] or a [[function field]] in one variable over a [[finite field]] and let ''E'' be an [[elliptic curve]]. If ''v'' is a [[Algebraic number field#Nonarchimedian or ultrametric places|non-archimedean place]] of [[Field norm|norm]] ''q<sub>v</sub>'' of ''K'' and a ∈ K, with ''v''(''a'') = 0 then ''v''(a<sup>''q<sub>v</sub>''-1</sup>-1) ≥ 1. ''v'' is called a ''Wieferich place'' for base ''a'', if ''v''(a<sup>''q<sub>v</sub>''-1</sup>-1) > 1, an ''elliptic Wieferich place'' for base ''P'' ∈ ''E'', if ''N<sub>v</sub>P'' ∈ ''E''<sub>2</sub> and a ''strong elliptic Wieferich place'' for base ''P'' ∈ ''E'' if ''n<sub>v</sub>P'' ∈ ''E''<sub>2</sub>, where ''n<sub>v</sub>'' is the order of ''P'' modulo ''v'' and ''N<sub>v</sub>'' gives the number of [[rational point]]s (over the [[residue field]] of ''v'') of the reduction of ''E'' at ''v''.<ref>{{Citation | first = J. F. | last = Voloch | title = Elliptic Wieferich Primes | journal = Journal of Number Theory | volume = 81 | year = 2000 | pages = 205–209 | doi = 10.1006/jnth.1999.2471}}</ref>{{rp|206}}
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| == See also ==
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| * [[Wolstenholme prime]] – another type of prime number which in the broadest sense also resulted from the study of FLT
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| * [[Table of congruences]] - lists other congruences satisfied by prime numbers
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| * [[PrimeGrid]] - primes search project
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| == References ==
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| {{Reflist|colwidth=30em}}
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| ==Further reading==
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| * {{citation | first=R. | last=Haussner | title=Über die Kongruenzen 2<sup>''p''-1</sup>-1 ≡ 0 (mod ''p''<sup>2</sup>) für die Primzahlen ''p''=1093 und 3511 | journal=Archiv for Mathematik og Naturvidenskab | volume=39 | issue=5 | year=1926 | language=German | page=7 | id=[[German National Library|DNB]] [http://d-nb.info/363953469/about/html 363953469] | jfm=52.0141.06 | url=http://jfm.sub.uni-goettingen.de/cgi-bin/jfmen/JFM/en/quick.html?first=1&maxdocs=20&type=html&an=JFM%2052.0141.06&format=complete}}
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| * {{citation | first=R. | last=Haussner | title=Über numerische Lösungen der Kongruenz ''u''<sup>''p''-1</sup>-1 ≡ 0 (mod ''p''<sup>2</sup>) | journal=Journal für die reine und angewandte Mathematik (Crelle's Journal). | volume=1927 | issue=156 | year=1927 | language=German | pages=223–226 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002169924 | doi=10.1515/crll.1927.156.223}}
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| * {{citation | first=P. |last=Ribenboim | authorlink=Paulo Ribenboim | title=Thirteen lectures on Fermat's Last Theorem | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90432-8 | pages=139, 151 }}
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| * {{citation |first=Richard K. |last=Guy |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |edition=3rd |publisher=[[Springer Verlag]] |year=2004 |isbn=0-387-20860-7 |page=14 }}
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| * {{citation | first1=R. E. | last1=Crandall | first2=C. | last2=Pomerance | title=Prime numbers: a computational perspective | publisher=Springer Science+Business Media, Inc. | year=2005 | pages=31–32 | isbn=0-387-25282-7 | url=http://thales.doa.fmph.uniba.sk/macaj/skola/teoriapoli/primes.pdf}}
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| * {{citation | first=P. | last=Ribenboim | title=The new book of prime number records | publisher=Springer-Verlag New York, Inc. | year=1996 | pages=333–346 | isbn=0-387-94457-5 | url=http://books.google.com/?id=72eg8bFw40kC&pg=PA333}}
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| == External links ==
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| * {{MathWorld |urlname=WieferichPrime |title=Wieferich prime}}
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| * [http://go.helms-net.de/math/expdioph/fermatquotients.pdf Fermat-/Euler-quotients (''a''<sup>''p''−1</sup> − 1)/''p''<sup>''k''</sup> with arbitrary ''k'']
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| * [http://library.uwinnipeg.ca/people/dobson/mathematics/Wieferich_primes.html A note on the two known Wieferich primes]
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| * PrimeGrid's [http://www.primegrid.com/forum_thread.php?id=3008&nowrap=true#45945 Wieferich Prime Search project] page
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| {{Prime number classes|state=collapsed}}
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| {{DEFAULTSORT:Wieferich Prime}}
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| [[Category:Classes of prime numbers]]
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| [[Category:Unsolved problems in mathematics]]
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