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Brown, William D.

doi:10.21236/ada403499

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2004

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“…Especially for a = 2, where only the two celebrated solutions p = 1093 and p = 3511 are known, the search has been pushed to considerably high limits, more recently in [6] up to 4 · 10 12 , and then extended to 8 · 10 12 by R. McIntosh [14] and to 4.8 · 10 13 by R. Brown [5]. Finally, an Internet based search conducted by J. Knauer and the second author [13] attained the limit of 1.25 · 10 15 .…”

Section: Introductionmentioning

confidence: 99%

Solutions of the congruence 𝑎^{𝑝-1}≡1 (mod 𝑝^{𝑟})

Keller¹,

Richstein

2004

Math. Comp.

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To supplement existing data, solutions of a p−1 ≡ 1 (mod p 2 ) are tabulated for primes a, p with 100 < a < 1000 and 10 4 < p < 10 11 . For a < 100, five new solutions p > 2 32 are presented. One of these, p = 188748146801 for a = 5, also satisfies the "reverse" congruence p a−1 ≡ 1 (mod a 2 ). An effective procedure for searching for such "double solutions" is described and applied to the range a < 10 6 , p < max (10 11 , a 2 ). Previous to this, congruences a p−1 ≡ 1 (mod p r ) are generally considered for any r ≥ 2 and fixed prime p to see where the smallest prime solution a occurs.

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Section: Introductionmentioning

confidence: 99%