Carmichael's totient function conjecture

In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), 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. The current lower bound for a counterexample to the Conjecture is 10¹⁰¹⁰, which was determined by Kevin Ford in 1998. Ford went on to prove that if there exists a counterexample to the Conjecture, then a positive fraction (that is infinitely many) of the integers are likewise counterexamples.

Although the conjecture is widely believed, Carl Pomerance gave a sufficient condition for an integer n to be a counterexample to the conjecture. According to this condition, n is a counterexample if for every prime p such that p − 1 divides φ(n), p² divides n. However Pomerance showed that the existence of such an integer is highly improbable.

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.

• Schlafly, A.; Wagon, S. (1994), "Carmichael's conjecture on the Euler function is valid below 10^10,000,000", Mathematics of Computation, 63 (207): 415–419, doi:10.2307/2153585, MR 1226815.

External links

• Weisstein, Eric W. "Carmichael's Totient Function Conjecture". MathWorld.