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).

1093, 1006003, 40487

Offset: 1

Author

Felix Fröhlich, Jul 07 2015

Keywords

• nonn
• hard
• bref
• more

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).

Examples

			a(1) = A001220(1) = 1093.
a(2) = A014127(2) = 1006003.
a(3) = A123692(3) = 40487.
		

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.

Crossrefs

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

Programs

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