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%I #27 Jan 03 2014 01:37:46
%S 0,1,4,5,16,20,64,65,80,81,85,145,256,260,320,324,325,340,405,580,625,
%T 629,689,949,1024,1025,1040,1105,1280,1296,1300,1360,1620,1649,2320,
%U 2401,2405,2465,2500,2501,2516,2581,2725,2756,3125,3425,3796,4096,4100,4160
%N Numbers of the form x^4 + 4*y^4.
%C Since 4 is even, either x or y or both may be negative integers, because their fourth powers will then be positive.
%C The only prime term in this sequence is 5; this can be proved using Sophie Germain's identity.
%D Titu Andreescu and Rǎzvan Gelca, Mathematical Olympiad Challenges, New York, Birkhäuser (2009), p. 48.
%H Charles R Greathouse IV, <a href="/A227855/b227855.txt">Table of n, a(n) for n = 1..10000</a>
%H Graeme Taylor, <a href="http://aleph.straylight.co.uk/sophie.pdf">Identity of Sophie Germain</a>, April 1, 2006.
%F x^4 + 4y^4 = (x^2 - 2xy + 2y^2)(x^2 + 2xy + 2y^2). This is Sophie Germain's identity.
%e 80 = 2^4 + 4 * 2^4.
%e 81 = 3^4 + 4 * 0^4.
%e 85 = 3^4 + 4 * 1^4.
%t 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 *)
%o (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
%Y Subsequences include A000583, A141046 and A001589.
%K nonn,easy
%O 1,3
%A _Alonso del Arte_, Oct 31 2013
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