A033235 Primes of the form x^2 + 55*y^2.

59, 71, 199, 229, 251, 269, 311, 379, 389, 499, 509, 631, 661, 691, 751, 839, 881, 929, 1049, 1061, 1171, 1181, 1279, 1321, 1409, 1439, 1499, 1571, 1609, 1699, 1721, 1741, 1901, 1951, 2029, 2069, 2269

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also primes of the form x^2 - xy + 14y^2 with x and y nonnegative. - T. D. Noe, May 08 2005

From Lechoslaw Ratajczak, Apr 09 2017: (Start)

Conjecture: consecutive elements of this sequence are consecutive primes satisfying the congruence b(k) == 1 (mod k) for k>0, where b(k) is recursive sequence defined as follows: b(k) = -b(k-1) - b(k-2) + b(k-3) - b(k-4) with b(0)=2, b(1)=1, b(2)=0, b(3)=-1.

(b(59) - 1) mod 59 = (-496870918 - 1) mod 59 = 0, 59 = a(1).

(b(71) - 1) mod 71 = (88081764473 - 1) mod 71 = 0, 71 = a(2).

For 10^6 consecutive positive integers there are 9748 prime solutions and 5 nonprime (1, 586, 2935, 17161, 429737) solutions of the congruence. (End)

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Programs

Mathematica

QuadPrimes2[1, 0, 55, 10000] (* see A106856 *)