A092572 Numbers of the form x^2 + 3y^2 where x and y are positive integers.

4, 7, 12, 13, 16, 19, 21, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 76, 79, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Author

Eric W. Weisstein, Feb 28 2004

Keywords

• nonn

Comments

Superset of primes of the form 6n+1 (A002476).

It seems that all integer solutions of ((a+b)^3 - (a-b)^3) / (2*b) = c^3 have c = x^2 + 3*y^2. - Juergen Buchmueller (pullmoll(AT)t-online.de), Apr 04 2008

To prove the case of cubes in Fermat's last theorem, Euler considered numbers of the form a^2 + 3b^2. In the equation x^3 + y^3 = z^3, Euler specified that x = a - b and y = a + b. - Alonso del Arte, Jul 19 2012

All terms == 0,1,3,4, or 7 (mod 9). - Robert Israel, Apr 03 2017

Examples

			7 is of the specified form, since 2^2 + 3 * 1^2 = 7.
So is 12, since 3^2 + 3 * 1^2 = 12, and 13, with 1^2 + 3 * 2^2 = 13.
		

References

• Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): 4.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
• Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Crossrefs

Cf. A002476, A092573, A092575, A158937 (similar definition but with duplicates left in).

Programs

Maple

N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2 + 3*y^2, x = 1 .. floor(sqrt(N - 3*y^2))),
  y=1..floor(sqrt(N/3-1)))}:
sort(convert(S,list)); # Robert Israel, Apr 03 2017

Mathematica

Union[Flatten[Table[a^2 + 3b^2, {a, 20}, {b, Ceiling[Sqrt[(400 - a^2)/3]]}]]] (* Alonso del Arte, Jul 19 2012 *)