%I #17 May 05 2024 08:57:04

%S 5,11,13,181,1523,1741,2521,19531,24421,29789,76543,108529,489061,

%T 880301,1769069,6811741

%N Split primes p such that prime P lying above p is a Wieferich place of K (with discriminant D_K), for some imaginary quadratic field K of class number 1.

%H D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands, <a href="https://doi.org/10.1016/S0022-314X(05)80027-7">Computation of Iwasawa Lambda invariants for imaginary quadratic fields</a>, Journal of Number Theory, Vol. 37, No. 1 (1991), 100-121.

%H Á. Lozano-Robledo, <a href="http://dx.doi.org/10.1016/j.jnt.2009.10.013">Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations</a>, Journal of Number Theory, Vol. 130, No. 3 (2010), 539-558. See table 2 on page 555.

%o (Sage)

%o def is_A275118(k):

%o if not Integer(k).is_prime(): return False

%o for D in [1, 2, 3, 7, 11, 19, 43, 67, 163]:

%o fct = QuadraticField(-D).ideal(k).factor()

%o if len(fct)==2:

%o pi = fct[1][0].gens_reduced()[0]

%o if (pi^(k-1) - 1).valuation(fct[0][0]) > 1: return True

%o return False

%o print([k for k in range(10^7) if is_A275118(k)]) # _Robin Visser_, Apr 26 2024

%Y Cf. A239902.

%K nonn,more

%O 1,1

%A _Felix Fröhlich_, Jul 18 2016

%E a(11)-a(16) from _Robin Visser_, Apr 26 2024