A218697 Semiprimes that can be written in the form x^2 + 5*y^2 with x, y > 0.

6, 9, 14, 21, 46, 49, 69, 86, 94, 129, 134, 141, 145, 161, 166, 201, 205, 206, 214, 249, 254, 301, 305, 309, 321, 326, 329, 334, 381, 445, 446, 454, 469, 489, 501, 505, 526, 529, 545, 566, 581, 614, 669, 681, 694, 721, 734, 745, 749, 766, 789, 841, 849, 886, 889

Offset: 1

Author

Arkadiusz Wesolowski, Nov 04 2012

Keywords

• nonn

Comments

If two primes which end in 3 or 7 and surpass by 3 a multiple of 4 are multiplied, then their product will be composed of a square and the quintuple of another square. (Fermat (1654))

Examples

			94 = 7^2 + 5*3^2, therefore 94 is a term.
		

References

• Dedekind R., Theory of Algebraic Integers, Cambridge Univ. Press, 1996 (translation of the 1877 French original), pp. 12-13.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A154778.

Programs

Mathematica

n = 889; limx = Sqrt[n]; limy = Sqrt[n/5]; Select[Union@Flatten@Table[x^2 + 5*y^2, {x, limx}, {y, limy}], # <= n && PrimeOmega[#] == 2 &]
Select[Select[Range[889], PrimeOmega[#] == 2 &], Length@FindInstance[y > 0 && x^2 + 5*y^2 == #, {x, y}, Integers] > 0 &] (* Arkadiusz Wesolowski, Jan 13 2013 *)

Formula

A154778 INTERSECT A001358.