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A275118

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.

5, 11, 13, 181, 1523, 1741, 2521, 19531, 24421, 29789, 76543, 108529, 489061, 880301, 1769069, 6811741

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OFFSET

1,1

LINKS

Table of n, a(n) for n=1..16.

D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands, Computation of Iwasawa Lambda invariants for imaginary quadratic fields, Journal of Number Theory, Vol. 37, No. 1 (1991), 100-121.

Á. Lozano-Robledo, Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations, Journal of Number Theory, Vol. 130, No. 3 (2010), 539-558. See table 2 on page 555.

PROG

(SageMath)

def is_A275118(k):

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

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

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

if len(fct)==2:

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

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

return False

print([k for k in range(10^7) if is_A275118(k)]) # Robin Visser, Apr 26 2024