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

Offset: 1

Author

Felix Fröhlich, Jul 18 2016

Keywords

• nonn
• more

Links

• 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.

Crossrefs

Cf. A239902.

Programs

Sage

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

Extensions

a(11)-a(16) from Robin Visser, Apr 26 2024