A139643 Primes of the form x^2+Ny^2, with N=102.

103, 127, 151, 223, 271, 409, 433, 457, 463, 577, 631, 727, 769, 919, 937, 967, 1033, 1039, 1063, 1087, 1249, 1279, 1327, 1447, 1471, 1543, 1657, 1753, 1759, 1777, 1783, 1801, 1879, 1951, 1993, 2089, 2143, 2161, 2287, 2311, 2473, 2503, 2551

Offset: 1

Author

T. D. Noe, Apr 29 2008

Keywords

• nonn
• easy

Comments

Discriminant=-408. N is an idoneal number (A000926), which means that the quadratic form's genus consists of a single class, which means that the primes of this form are identical to the primes that are congruent to c (mod 4N), where c is a set of numbers less than 4N. The sequence A139642 lists the set c for each idoneal number. That sequence also cross references the sequences for the quadratic forms with N equal to the first 36 idoneal numbers. The remaining quadratic forms are this sequence and the 28 listed in order below. Note that the sequences for N=120 and 240 are the same.

The primes are congruent to {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361} (mod 408).

References

• David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
• L. E. Dickson, History of the Theory of Numbers, Vol 3, Chelsea, 1923.

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)

Crossrefs

Cf. A139644, A139645, A139502, A139646, A139647, A139648, A139506, A139649, A139650, A139651, A139652, A139502, A139653-A139668.

Programs

Magma

[ p: p in PrimesUpTo(3000) | p mod 408 in {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361}]; // Vincenzo Librandi, Jul 28 2012

Magma

k:=102; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

Maple

C:= [1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361]:
select(isprime, [seq(seq(408*i+j,j=C),i=0..100)]); # Robert Israel, Jul 03 2016

Mathematica

nn=102; pMax=10000; Union[Reap[Do[p=x^2+nn*y^2; If[p<=pMax && PrimeQ[p], Sow[p]], {x,Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2,1]]] (* T. D. Noe, Aug 02 2009 *)
QuadPrimes2[1, 0, 102, 10000] (* see A106856 *)