Carmichael's totient function conjecture: Difference between revisions

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[[Robert Daniel Carmichael|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 10^10⁷ by Schlafly and Wagon. The current lower bound for a counterexample to the Conjecture is 10^10¹⁰, which was determined by Kevin Ford in 1998.

Although the conjecture is widely believed, Carl Pomerance gave a sufficient condition for an integer ''n'' that can be a counterexample to the conjecture. This condition states that the natural number (or integer) ''n'' is a counterexample if for every prime ''p'', (''p -1'') divides &phi;(''n'')implies ''p''<sup>2 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