A138760 Numbers n such that n^4 is a sum of 4th powers of four nonzero integers whose sum is n.

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

Offset: 1

Author

Jonathan Sondow, Mar 28 2008

Keywords

• nonn

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).

Examples

			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).
		

Links

• 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

Crossrefs

Cf. A003294, A003828, A096739.

Formula

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