

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.


2



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
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OFFSET

1,1


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



MATHEMATICA

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


PROG

(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.


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



