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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 (list; graph; refs; listen; history; text; internal format)
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) 1027-1028.

Noam Elkies, On A^4 + B^4 + C^4 = D^4, Math. Comp. 51 (1988) 825-835.

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) 220-236.

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 (Jacobi-Madden).

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

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Last modified April 13 10:24 EDT 2021. Contains 342935 sequences. (Running on oeis4.)