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A259075 Smallest base b > 1 such that both prime(n) and prime(n+1) are base-b Wieferich primes, i.e., p = prime(n) satisfies  b^(p-1) == 1 (mod p^2) and q = prime(n+1) satisfies b^(q-1) == 1 (mod q^2). 2

%I

%S 17,26,18,148,239,249,423,28,63,374,117,787,2059,1085,655,4586,4153,

%T 3147,10056,4559,2092,18692,19487,3018,19343,14285,164,31469,6817,

%U 7916,16128,4505,18768,2752,26664,16717,129702,46171,1040,3608,9479,4840,42348,14128

%N Smallest base b > 1 such that both prime(n) and prime(n+1) are base-b Wieferich primes, i.e., p = prime(n) satisfies b^(p-1) == 1 (mod p^2) and q = prime(n+1) satisfies b^(q-1) == 1 (mod q^2).

%C Does b exist for all n?

%C a(n) == A039678(n) iff A039678(n) == A039678(n+1). The smallest n where those equalities hold is n = 8.

%H Felix Fröhlich, <a href="/A259075/b259075.txt">Table of n, a(n) for n = 1..1000</a>

%t a[n_] := Block[{b=2, p = Prime@{n, n+1}}, While[{1,1} != PowerMod[ b, p-1, p^2], b++]; b]; Array[a, 40] (* _Giovanni Resta_, Jun 23 2015 *)

%o (PARI) a(n) = p=prime(n); q=prime(n+1); b=2; while(Mod(b, p^2)^(p-1)!=1 || Mod(b, q^2)^(q-1)!=1, b++); b

%Y Cf. A039678.

%K nonn

%O 1,1

%A _Felix Fröhlich_, Jun 18 2015

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Last modified November 22 00:32 EST 2019. Contains 329383 sequences. (Running on oeis4.)