A323271 Numbers of the form p*q*r where p, q, r are distinct primes congruent to 1 mod 4 such that Legendre(p/q) = Legendre(p/r) = Legendre(q/r) = -1.

2405, 3145, 4745, 6205, 6305, 8245, 8905, 9605, 12545, 12805, 14705, 16405, 16745, 17945, 18241, 19045, 19345, 19805, 20213, 20605, 20905, 22945, 23545, 25805, 26605, 26945, 28645, 29705, 30073, 33745, 35705, 35989, 36205, 36305, 37505, 38369, 38545

Offset: 1

Author

N. J. A. Sloane, Jan 11 2019

Keywords

• nonn

Comments

If k is a term, the Pell equation x^2 - k*y^2 = -1 has a solution [Dirichlet, Newman (1977)]. This is only a sufficient condition, there are many other solutions, see A031396.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Morris Newman, A note on an equation related to the Pell equation, The American Mathematical Monthly 84.5 (1977): 365-366.

Crossrefs

Cf. A002144, A031396, A322781, A323272.

Programs

Python

from sympy.ntheory import legendre_symbol, factorint
A323271_list, k = [], 1
while len(A323271_list) < 10000:
    fk, fv = zip(*list(factorint(4*k+1).items()))
    if sum(fv) == len(fk) == 3 and fk[0] % 4 == fk[1] % 4 == fk[2] % 4 == 1 and legendre_symbol(fk[0],fk[1]) == legendre_symbol(fk[0],fk[2]) == legendre_symbol(fk[1],fk[2]) == -1:
            A323271_list.append(4*k+1)
    k += 1 # Chai Wah Wu, Jan 11 2019