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{{Short description|Problem in number theory on equal totients}}
In mathematics, '''Carmichael's totient function conjecture''' concerns the [[Multiplicity (mathematics)|multiplicity]] of values of [[Euler's totient function]] ''φ''(''n''), which counts the number of integers less than and [[coprime]] to ''n''. It states that, for every ''n'' there is at least one other integer ''m'' ≠ ''n'' such that ''φ''(''m'') = ''φ''(''n'').
[[Robert Daniel Carmichael|Robert Carmichael]] first stated this [[Conjecture (mathematics)|conjecture]] in 1907, but as a [[theorem]] rather than as a conjecture. However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an [[open problem]].
==Examples==
The totient function ''φ''(''n'') is equal to 2 when ''n'' is one of the three values 3, 4, and 6. Thus, if we take any one of these three values as ''n'', then either of the other two values can be used as the ''m'' for which ''φ''(''m'') = ''φ''(''n'').
Similarly, the totient is equal to 4 when ''n'' is one of the four values 5, 8, 10, and 12, and it is equal to 6 when ''n'' is one of the four values 7, 9, 14, and 18. In each case, there is more than one value of ''n'' having the same value of ''φ''(''n'').
The conjecture states that this phenomenon of repeated values holds for every ''n''.
{|class="wikitable"
|''k''
|numbers ''n'' such that ''φ''(''n'') = ''k'' {{OEIS|id=A032447}}
|number of such ''n''s {{OEIS|id=A014197}}
|-
|1
|1, 2
|2
|-
|2
|3, 4, 6
|3
|-
|4
|5, 8, 10, 12
|4
|-
|6
|7, 9, 14, 18
|4
|-
|8
|15, 16, 20, 24, 30
|5
|-
|10
|11, 22
|2
|-
|12
|13, 21, 26, 28, 36, 42
|6
|-
|16
|17, 32, 34, 40, 48, 60
|6
|-
|18
|19, 27, 38, 54
|4
|-
|20
|25, 33, 44, 50, 66
|5
|-
|22
|23, 46
|2
|-
|24
|35, 39, 45, 52, 56, 70, 72, 78, 84, 90
|10
|-
|28
|29, 58
|2
|-
|30
|31, 62
|2
|-
|32
|51, 64, 68, 80, 96, 102, 120
|7
|-
|36
|37, 57, 63, 74, 76, 108, 114, 126
|8
|-
|40
|41, 55, 75, 82, 88, 100, 110, 132, 150
|9
|-
|42
|43, 49, 86, 98
|4
|-
|44
|69, 92, 138
|3
|-
|46
|47, 94
|2
|-
|48
|65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210
|11
|-
|52
|53, 106
|2
|-
|54
|81, 162
|2
|-
|56
|87, 116, 174
|3
|-
|58
|59, 118
|2
|-
|60
|61, 77, 93, 99, 122, 124, 154, 186, 198
|9
|-
|64
|85, 128, 136, 160, 170, 192, 204, 240
|8
|-
|66
|67, 134
|2
|-
|70
|71, 142
|2
|-
|72
|73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270
|17
|}
==Lower bounds==
There are very high [[lower bound]]s for Carmichael's conjecture that are relatively easy to determine. Carmichael himself proved that any counterexample to his conjecture (that is, a value ''n'' such that φ(''n'') is different from the totients of all other numbers) must be at least 10<sup>37</sup>, and [[Victor Klee]] extended this result to 10<sup>400</sup>. A lower bound of <math>10^{10^7}</math>was given by Schlafly and Wagon, and a lower bound of <math>10^{10^{10}}</math> was determined by [[Kevin Ford (mathematician)|Kevin Ford]] in 1998.<ref name=HBII228>Sándor & Crstici (2004) p. 228</ref>
The computational technique underlying these lower bounds depends on some key results of Klee that make it possible to show that the smallest counterexample must be divisible by squares of the primes dividing its totient value. Klee's results imply that 8 and Fermat primes (primes of the form 2<sup>''k''</sup>
==Other results==
Ford also proved that if there exists a counterexample to the
Although the conjecture is widely believed, [[Carl Pomerance]] gave a sufficient condition for an integer ''n'' to be a counterexample to the conjecture {{harv|Pomerance|1974}}. According to this condition, ''n'' is a counterexample if for every prime ''p'' such that ''p'' − 1 divides ''φ''(''n''), ''p''<sup>2</sup> divides
Another way of stating Carmichael's conjecture is that, if
''A''(''f'') denotes the number of positive integers ''n'' for which ''φ''(''n'') = ''f'', then ''A''(''f'') can never equal
==Notes==
Line 32 ⟶ 160:
| journal = [[Bulletin of the American Mathematical Society]]
| pages = 241–243
| title = On Euler's ''φ''-function
| volume = 13
| year = 1907
| mr = 1558451
}}.
*{{citation
| last = Carmichael | first = R. D. | author-link = Robert Daniel Carmichael
Line 43 ⟶ 171:
| journal = [[Bulletin of the American Mathematical Society]]
| pages = 109–110
| title = Note on Euler's ''φ''-function
| volume = 28
| year = 1922
| mr = 1560520
}}.
*{{citation
| last = Ford | first = K. | author-link = Kevin Ford (mathematician)
| doi = 10.2307/121103
| issue = 1
| journal = [[Annals of Mathematics]]
| pages = 283–311
| title = The number of solutions of ''φ''(''x'') = ''m''
| volume = 150
| year = 1999
Line 60 ⟶ 189:
* {{citation |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 | at=B39 }}.
*{{citation
| last = Klee | first = V. L.
| doi = 10.1090/S0002-9904-1947-08940-0
| journal = [[Bulletin of the American Mathematical Society]]
Line 68 ⟶ 197:
| year = 1947
| mr = 0022855 | zbl=0035.02601
| issue = 12
}}.
*{{citation
| last1 = Pomerance | first1 = Carl | author-link = Carl Pomerance
Line 79 ⟶ 209:
| zbl=0254.10009
| url = http://www.math.dartmouth.edu/~carlp/PDF/carmichaelconjecture.pdf
| doi=10.2307/2038881| jstor = 2038881 }}.
* {{citation | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | ___location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=978-1-4020-2546-
*{{citation
| last1 = Schlafly | first1 = A.
Line 95 ⟶ 225:
==External links==
*{{mathworld|title=Carmichael's Totient Function Conjecture|urlname=CarmichaelsTotientFunctionConjecture|mode=cs2}}
[[Category:Multiplicative functions]]
[[Category:Conjectures]]
[[Category:Unsolved problems in number theory]]
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