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Wieferich primes to base 26.
4

%I #10 Aug 23 2024 17:23:50

%S 3,5,71,486999673,6695256707

%N Wieferich primes to base 26.

%C Prime numbers p such that p^2 divides 26^(p-1) - 1.

%C No more terms up to 9.8*10^13.

%H Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/">Fermatquotient B^(P-1) == 1 (mod P^2)</a>

%H P. L. Montgomery, <a href="http://www.jstor.org/stable/2152960">New Solutions of a^p-1 == 1 (mod p^2)</a>, Mathematics of Computation, Vol. 61, No. 203 (1993), 361-363.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wieferich_prime">Wieferich prime</a>

%t Select[Prime[Range[26*10^6]],PowerMod[26,#-1,#^2]==1&] (* The program generates the first 4 terms of the sequence. *) (* _Harvey P. Dale_, Aug 23 2024 *)

%o (PARI) forprime(p=2, , if(Mod(26, p^2)^(p-1)==1, print1(p, ", ")))

%Y Wieferich primes to base b: A001220 (b=2), A014127 (b=3), A123692 (b=5), A212583 (b=6), A123693 (b=7), A045616 (b=10), A111027 (b=12), A128667 (b=13), A234810 (b=14), A242741 (b=15), A128668 (b=17), A244260 (b=18), A090968 (b=19), A242982 (b=20), A298951 (b=22), A128669 (b=23), this sequence (b=26), A306256 (b=30).

%K nonn,hard,more

%O 1,1

%A _Jianing Song_, Feb 01 2019