A003294 Numbers k such that k^4 can be written as a sum of four positive 4th powers.
353, 651, 706, 1059, 1302, 1412, 1765, 1953, 2118, 2471, 2487, 2501, 2604, 2824, 2829, 3177, 3255, 3530, 3723, 3883, 3906, 3973, 4236, 4267, 4333, 4449, 4557, 4589, 4942, 4949, 4974, 5002, 5208, 5281, 5295, 5463, 5491, 5543, 5648, 5658
Offset: 1
Examples
353^4 = 30^4 + 120^4 + 272^4 + 315^4. 651^4 = 240^4 + 340^4 + 430^4 + 599^4. 2487^4 = 435^4 + 710^4 + 1384^4 + 2420^4. 2501^4 = 1130^4 + 1190^4 + 1432^4 + 2365^4. 2829^4 = 850^4 + 1010^4 + 1546^4 + 2745^4.
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- D. Wells, Curious and interesting numbers, Penguin Books, p. 139.
Links
- T. D. Noe, Table of n, a(n) for n = 1..4870 (using Wroblewski's results)
- Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.
- 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.
- Kermit Rose and Simcha Brudno, More about four biquadrates equal one biquadrate, Math. Comp., 27 (1973), 491-494.
- Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.
- Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000.
- Index to sequences related to diophantine equations (4,1,4).
Programs
-
Mathematica
fourthPowerSums = {}; Do[a4 = a^4; Do[b4 = b^4; Do[c4 = c^4; Do[d4 = d^4; e4 = a4 + b4 + c4 + d4; e = Sqrt[Sqrt[e4]]; If[IntegerQ[e], AppendTo[fourthPowerSums, e]], {d, c + 1, 9000}], {c, b + 1, 6000}],{b, a + 1, 5000}], {a, 30, 3000}]; Union @ fourthPowerSums (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)
Extensions
Corrected and extended by Don Reble, Jul 07 2007
More terms from David Wasserman, Nov 16 2007
Definition clarified by Jonathan Sondow, Apr 06 2008
Comments