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A227855
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Numbers of the form x^4 + 4*y^4.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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Titu Andreescu and Rǎzvan Gelca, Mathematical Olympiad Challenges, New York, Birkhäuser (2009), p. 48.
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LINKS
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FORMULA
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x^4 + 4y^4 = (x^2 - 2xy + 2y^2)(x^2 + 2xy + 2y^2). This is Sophie Germain's identity.
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EXAMPLE
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80 = 2^4 + 4 * 2^4.
81 = 3^4 + 4 * 0^4.
85 = 3^4 + 4 * 1^4.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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