A259909 - OEIS

%I #29 Jul 15 2015 21:46:53

%S 1093,1006003,40487

%N n-th Wieferich prime to base prime(n), i.e., primes p such that p is the n-th solution of the congruence (prime(n))^(p-1) == 1 (mod p^2).

%C Main diagonal of table T(b, p) of Wieferich primes p to prime bases b (that table is not yet in the OEIS as a sequence).

%C a(4), if it exists, corresponds to A123693(4) and is larger than 9.7*10^14 (cf. Dorais, Klyve, 2011).

%C a(5), if it exists, corresponds to the 5th base-11 Wieferich prime and is larger than approximately 5.9*10^13 (cf. Fischer).

%C a(6), if it exists, corresponds to A128667(6) and is larger than approximately 5.9*10^13 (cf. Fischer).

%D W. Keller, Prime solutions p of a^p-1 = 1 (mod p2) for prime bases a, Abstracts Amer. Math. Soc., 19 (1998), 394.

%H M. Aaltonen and K. Inkeri, <a href="http://dx.doi.org/10.1090/S0025-5718-1991-1052082-7">Catalan's equation x^p - y^q and related congruences</a>, Mathematics of Computation, Vol. 56 No. 193 (1991), 359-370.

%H F. G. Dorais and D. Klyve, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html">A Wieferich prime search up to p < 6.7*10^15</a>, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.

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

%H W. Keller and J. Richstein, <a href="http://web.archive.org/web/20091109011757/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html">Fermat quotients q_p(a) that are divisible by p</a> (Cached copy at the Wayback Machine).

%H K. E. Kloss, <a href="http://dx.doi.org/10.6028/jres.069B.035">Some Number-Theoretic Calculations</a>, J. Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335-336.

%e a(1) = A001220(1) = 1093.

%e a(2) = A014127(2) = 1006003.

%e a(3) = A123692(3) = 40487.

%o (PARI) a(n) = my(i=0, p=2); while(i < n, if(Mod(prime(n), p^2)^(p-1)==1, i++; if(i==n, break({1}))); p=nextprime(p+1)); p

%Y Cf. A001220, A014127, A123692, A123693, A174422, A178871.

%K nonn,hard,bref,more

%O 1,1

%A _Felix Fröhlich_, Jul 07 2015