Carmichael's totient function conjecture: Difference between revisions

Revision as of 14:19, 16 September 2008

Revision as of 20:49, 16 September 2008

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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 canto be a counterexample to the conjecture. ThisAccording conditionto statesthis that the natural number (or integer)condition, ''n'' is a counterexample if for every prime ''p'', (such that ''p -1'') − 1 divides φ(''n'')implies, ''p''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