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In mathematics, a '''noncototient''' is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of [[coprime]] integers below it. That is, ''m'' − φ(''m'') = ''n'', where φ stands for [[Euler's totient function]], has no solution for ''m''.  The ''[[cototient]]'' of ''n'' is defined as ''n'' − φ(''n''), so a '''noncototient''' is a number that is never a cototient.


It is conjectured that all noncototients are even. This follows from a modified form of the [[Goldbach conjecture]]: if the even number ''n'' can be represented as a sum of two distinct primes ''p'' and ''q,'' then


: <math>
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pq - \varphi(pq) = pq - (p-1)(q-1) = p+q-1 = n-1. \,
</math>
 
It is expected that every even number larger than 6 is a sum of distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations <math>1=2-\phi(2), 3 = 9 - \phi(9)</math> and <math>5 = 25 - \phi(25)</math>.
 
The first few noncototients are:
 
[[10 (number)|10]], [[26 (number)|26]], [[34 (number)|34]], [[50 (number)|50]], [[52 (number)|52]], [[58 (number)|58]], [[86 (number)|86]], [[100 (number)|100]], [[116 (number)|116]], [[122 (number)|122]], [[130 (number)|130]], [[134 (number)|134]], [[146 (number)|146]], [[154 (number)|154]], [[170 (number)|170]], [[172 (number)|172]], 186, 202, 206, 218, [[222 (number)|222]], 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, 518, 520 {{OEIS|id=A005278}}
 
[[Paul Erdős|Erdős]] (1913-1996) and [[Wacław Sierpiński|Sierpinski]] (1882-1969) asked whether there exist infinitely many noncototients. This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family <math> 2^k \cdot 509203</math> is an example. Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca (2000).
 
==References==
* {{cite journal | zbl=0820.11003 | last1=Browkin | first1=J. | last2=Schinzel | first2=A. | title=On integers not of the form n-φ(n) | journal=Colloq. Math. | volume=68 | number=1 | pages=55–58 | year=1995 }}
* {{cite journal | zbl=0965.11003 | last1=Flammenkamp | first1=A. | last2=Luca | first2=F. | title=Infinite families of noncototients | journal=Colloq. Math. | volume=86 | number=1 | pages=37–41 | year=2000 }}
* {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | pages=138–142}}
 
== External links ==
* [http://mathworld.wolfram.com/Noncototient.html Noncototient definition from MathWorld]
 
{{Totient}}
{{Classes of natural numbers}}
 
[[Category:Integer sequences]]

Revision as of 10:19, 4 March 2014


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