Carmichael's Totient Function Conjecture

It is thought that the Totient Valence Function $N_\phi(m)\geq 2$ (i.e., the Totient Valence Function never takes the value 1). This assertion is called Carmichael's totient function conjecture and is equivalent to the statement that there exists an $m\not=n$ such that $\phi(n)=\phi(m)$ (Ribenboim 1996, pp. 39-40). Any counterexample to the conjecture must have more than 10,000 Digits (Conway and Guy 1996). Recently, the conjecture was reportedly proven by F. Saidak in November, 1997 with a proof short enough to fit on a postcard.

See also Totient Function, Totient Valence Function

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 155, 1996.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996.