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Computational Evidence for Lehmer's Totient Conjecture
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| Organization: | Wolfram Research, Inc. |
| Department: | Information Resources |
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2004-11-23
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In 1932, D. H. Lehmer considered the equation k ϕ[N] == N - 1. for positive integers N and k. If N is prime, then the equation has a solution with k = 1. He conjectured that these are the only solutions.
Conjecture: Let a be a positive integer. Then ϕ[N] divides N - 1 if and only if N is prime.
The conjecture remains open to this day. The current best result, due to Cohen and Hagis, is that N must have at least 14 prime factors and be greater than 10^22. The code in this notebook carries out a search for a counter-example to Lehmer's conjecture and extends these limits.
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Lehmer Totient
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| LehmerArticle.nb (12.1 KB) - Mathematica Notebook [for Mathematica 5.0] |
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