1,1

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).

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

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

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).

Cf. A003294, A003828, A096739.

Sequence in context: A031572 A045108 A137816 this_sequence A045132 A096702 A165443

Adjacent sequences: A138757 A138758 A138759 this_sequence A138761 A138762 A138763

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Mar 28 2008