In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), the function which counts the number of integers less than and coprime to n.
This function φ(n) is equal to 2 when n is one of the three values 3, 4, and 6. It is equal to 4 when n is one of the four values 5, 8, 10, and 12. 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. That is, for every n there is at least one other integer m ≠ n such that φ(m) = φ(n). Robert Carmichael first stated this conjecture 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. Carmichael 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 1037, and Victor Klee extended this result to 10400. Carmichael's conjecture has since been verified computationally for all n less than or equal to 10107 by Schlafly and Wagon.
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 1. Relatedly, Wacław Sierpiński conjectured that every positive integer other than 1 occurs as a value of A(f), a conjecture that was proven in 1999 by Kevin Ford.
References
- Carmichael, R. D. (1907), "On Euler's φ-function", Bulletin of the American Mathematical Society, 13 (5): 241–243, doi:10.1090/S0002-9904-1907-01453-2, MR 1558451.
- Carmichael, R. D. (1922), "Note on Euler's φ-function", Bulletin of the American Mathematical Society, 28 (3): 109–110, doi:10.1090/S0002-9904-1922-03504-5, MR 1560520.
- Ford, K. (1999), "The number of solutions of φ(x) = m", Annals of Mathematics, 150 (1): 283–311, doi:10.2307/121103, MR 1715326.
- Klee, V. L., Jr. (1947), "On a conjecture of Carmichael", Bulletin of the American Mathematical Society, 53: 1183–1186, doi:10.1090/S0002-9904-1947-08940-0, MR 0022855
{{citation}}: CS1 maint: multiple names: authors list (link).
- Schlafly, A.; Wagon, S. (1994), "Carmichael's conjecture on the Euler function is valid below 1010,000,000", Mathematics of Computation, 63 (207): 415–419, doi:10.2307/2153585, MR 1226815.
External links
 | This number theory-related article is a stub. You can help Wikipedia by expanding it. |