

A138760


Numbers n such that n^4 is a sum of 4th powers of four nonzero integers whose sum is n.


1



5491, 10982, 16473, 21964, 27455, 32946, 38437, 43928, 49419, 51361, 54910, 60401, 65892, 71383, 76874, 82365, 87856, 93347, 98838, 102722, 104329, 109820, 115311, 120802, 126293, 131784, 137275, 142766, 148257, 153748, 154083, 159239, 164730
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OFFSET

1,1


COMMENTS

Any multiple of a member is also a member. A member that is not a multiple of another member is called primitive. Using elliptic curves, Jacobi and Madden prove that there are infinitely many primitive members. According to them, the only primitive members less than 222,000 are 5491 (due to Brudno) and 51361 (due to Wroblewski).


LINKS

Table of n, a(n) for n=1..33.
Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 10271028.
Noam Elkies, On A^4 + B^4 + C^4 = D^4, Math. Comp. 51 (1988) 825835.
Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4, Amer. Math. Monthly 115 (2008) 220236.
Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4
Eric Weisstein's MathWorld, Diophantine equation  4th powers
Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000


FORMULA

n^4 = a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 with abcd =/= 0.


EXAMPLE

5491^4 = 5400^4 + (2634)^4 + 1770^4 + 955^4 and 5491 = 5400  2634 + 1770 + 955, so 5491 is a member (Brudno).
51361^4 = 48150^4 + (31764)^4 + 27385^4 + 7590^4 and 51361 = 48150  31764 + 27385 + 7590, so 51361 is a member (Wroblewski).
1347505009^4 = 1338058950^4 + (89913570)^4 + 504106884^4 + (404747255)^4, and 1347505009 = 1338058950  89913570 + 504106884  404747255, so 1347505009 is a member (JacobiMadden).


CROSSREFS

Cf. A003294, A003828, A096739.
Sequence in context: A232153 A203791 A137816 * A251885 A251878 A251877
Adjacent sequences: A138757 A138758 A138759 * A138761 A138762 A138763


KEYWORD

nonn


AUTHOR

Jonathan Sondow, Mar 28 2008


STATUS

approved



