A078116 Primes of the form x^2 + 2y^2 where y<=x. Terms are listed in increasing order of x; for fixed x they're in increasing order of y.

3, 11, 17, 43, 67, 83, 89, 113, 131, 179, 139, 193, 283, 241, 331, 457, 227, 233, 257, 353, 467, 563, 617, 307, 577, 739, 379, 433, 523, 811, 1009, 443, 449, 491, 569, 641, 683, 953, 1019, 1163, 547, 601, 691, 643, 787, 1777, 761, 827, 857, 929, 971, 1307

Offset: 1

Author

Cino Hilliard, Dec 05 2002

Keywords

• easy
• nonn

Comments

Every prime of the form 8n+1 or 8n+3 has a unique representation of the form x^2 + 2y^2 with positive integers x and y. This sequence has the primes for which y<=x.

References

• Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press 1972, p. 276 (Fermat prime number theorems).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Programs

Mathematica

Select[Flatten[Table[x^2+2y^2, {x, 0, 30}, {y, 0, x}]], PrimeQ]

PARI

sqplus2sq(n,m) = ct=0; for(x=1,n, for(y=1,x, s = x^2+m*y^2; if(isprime(s),ct+=1; print1(s" "); ); ); ); \\ Lists primes of the form x^2+m*y^2 with 1<=y<=x<=n.

Extensions

Edited by Dean Hickerson, Dec 12 2002