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 A259909 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). 1
 1093, 1006003, 40487 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). a(4), if it exists, corresponds to A123693(4) and is larger than 9.7*10^14 (cf. Dorais, Klyve, 2011). a(5), if it exists, corresponds to the 5th base-11 Wieferich prime and is larger than approximately 5.9*10^13 (cf. Fischer). a(6), if it exists, corresponds to A128667(6) and is larger than approximately 5.9*10^13 (cf. Fischer). REFERENCES W. Keller, Prime solutions p of a^p-1 = 1 (mod p2) for prime bases a, Abstracts Amer. Math. Soc., 19 (1998), 394. LINKS M. Aaltonen and K. Inkeri, Catalan's equation x^p - y^q and related congruences, Mathematics of Computation, Vol. 56 No. 193 (1991), 359-370. F. G. Dorais and D. Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14. R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2) W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p (Cached copy at the Wayback Machine). K. E. Kloss, Some Number-Theoretic Calculations, J. Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335-336. EXAMPLE a(1) = A001220(1) = 1093. a(2) = A014127(2) = 1006003. a(3) = A123692(3) = 40487. PROG (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 CROSSREFS Cf. A001220, A014127, A123692, A123693, A174422, A178871. Sequence in context: A307220 A091674 A022197 * A124122 A163561 A203807 Adjacent sequences:  A259906 A259907 A259908 * A259910 A259911 A259912 KEYWORD nonn,hard,bref,more AUTHOR Felix Fröhlich, Jul 07 2015 STATUS approved

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Last modified June 25 04:06 EDT 2021. Contains 345451 sequences. (Running on oeis4.)