Editing Euler's totient function (section)

==Unsolved problems==

===Lehmer's conjecture===

{{main article|Lehmer's totient problem}}

If {{mvar|p}} is prime, then {{math|''φ''(''p'') {{=}} ''p'' − 1}}. In 1932 [[D. H. Lehmer]] asked if there are any composite numbers {{mvar|n}} such that {{math|''φ''(''n'') }} divides {{math|''n'' − 1}}. None are known.<ref>Ribenboim, pp. 36–37.</ref>

In 1933 he proved that if any such {{mvar|n}} exists, it must be odd, square-free, and divisible by at least seven primes (i.e. {{math|''ω''(''n'') ≥ 7}}). In 1980 Cohen and Hagis proved that {{math|''n'' > 10<sup>20</sup>}} and that {{math|''ω''(''n'') ≥ 14}}.<ref>{{cite journal | zbl=0436.10002 | last1=Cohen | first1=Graeme L. | last2=Hagis | first2=Peter Jr. | title=On the number of prime factors of {{mvar|n}} if {{math|''φ''(''n'')}} divides {{math|''n'' − 1}} | journal=Nieuw Arch. Wiskd. |series=III Series | volume=28 | pages=177–185 | year=1980 | issn=0028-9825 }}</ref> Further, Hagis showed that if 3 divides {{mvar|n}} then {{math|''n'' > 10<sup>1937042</sup>}} and {{math|''ω''(''n'') ≥ 298848}}.<ref>{{cite journal | zbl=0668.10006 | last=Hagis | first=Peter Jr. | title=On the equation {{math|''M''·φ(''n'') {{=}} ''n'' − 1}} | journal=Nieuw Arch. Wiskd. |series=IV Series | volume=6 | number=3 | pages=255–261 | year=1988 | issn=0028-9825 }}</ref><ref name=Guy142>Guy (2004) p.142</ref>

===Carmichael's conjecture===

{{main article|Carmichael's totient function conjecture}}

This states that there is no number {{mvar|n}} with the property that for all other numbers {{mvar|m}}, {{math|''m'' ≠ ''n''}}, {{math|''φ''(''m'') ≠ ''φ''(''n'')}}. See [[#Ford's theorem|Ford's theorem]] above.

As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.<ref name=Guy144/>

===Riemann hypothesis===

The [[Riemann hypothesis]] is true if and only if the inequality
:<math>\frac{n}{\varphi (n)}<e^\gamma \log\log n+\frac{e^\gamma (4+\gamma-\log 4\pi)}{\sqrt{\log n}}</math>
is true for all {{math|''n'' ≥ ''p''<sub>120569</sub>#}} where {{mvar|γ}} is [[Euler's constant]] and {{math|''p''<sub>120569</sub>#}} is the [[Primorial|product of the first]] {{math|120569}} primes.<ref>{{Cite book |last1=Broughan |first1=Kevin |title=Equivalents of the Riemann Hypothesis, Volume One: Arithmetic Equivalents |publisher=Cambridge University Press |year=2017 |edition=First |isbn=978-1-107-19704-6}} Corollary 5.35</ref>