A227855 Numbers of the form x^4 + 4*y^4.

0, 1, 4, 5, 16, 20, 64, 65, 80, 81, 85, 145, 256, 260, 320, 324, 325, 340, 405, 580, 625, 629, 689, 949, 1024, 1025, 1040, 1105, 1280, 1296, 1300, 1360, 1620, 1649, 2320, 2401, 2405, 2465, 2500, 2501, 2516, 2581, 2725, 2756, 3125, 3425, 3796, 4096, 4100, 4160

Offset: 1

Author

Alonso del Arte, Oct 31 2013

Keywords

• nonn
• easy

Comments

Since 4 is even, either x or y or both may be negative integers, because their fourth powers will then be positive.

The only prime term in this sequence is 5; this can be proved using Sophie Germain's identity.

Examples

			80 = 2^4 + 4 * 2^4.
81 = 3^4 + 4 * 0^4.
85 = 3^4 + 4 * 1^4.
		

References

• Titu Andreescu and Rǎzvan Gelca, Mathematical Olympiad Challenges, New York, Birkhäuser (2009), p. 48.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Graeme Taylor, Identity of Sophie Germain, April 1, 2006.

Crossrefs

Subsequences include A000583, A141046 and A001589.

Programs

Mathematica

nn = 10; Union[Select[Flatten[Table[x^4 + 4*y^4, {x, 0, nn}, {y, 0, nn}]], # <= nn^4 &]] (* T. D. Noe, Nov 08 2013 *)

PARI

list(lim)=my(v=List(),t); for(y=0,sqrtnint(lim\4,4), for(x=0, sqrtnint(lim\1-(t=4*y^4),4), listput(v,t+x^4))); Set(v) \\ Charles R Greathouse IV, Nov 07 2013

Formula

x^4 + 4y^4 = (x^2 - 2xy + 2y^2)(x^2 + 2xy + 2y^2). This is Sophie Germain's identity.