A003828 Numbers k such that k^4 is a primitive sum of 3 positive fourth powers.

422481, 2813001, 8707481, 12197457, 16003017, 16430513, 20615673, 44310257, 68711097, 117112081, 145087793, 156646737, 589845921, 638523249, 873822121, 1259768473, 1679142729, 1787882337, 1871713857

Offset

1,1

Comments

There are no further terms up to 1986560000. - Robert Gerbicz, May 17 2009

References

R. K. Guy, Unsolved Problems in Number Theory, D1.

A. van der Poorten, Notes on Fermat's Last Theorem, Wiley, p. 46.

Links

Table of n, a(n) for n=1..19.

N. Elkies, On A^4 + B^4 + C^4 = D^4, Math. Comp. 51:184 (1988), 825-835.

D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d), Math. Comp. 70:233 (2001), 389-394.

Finished distributed computation for solutions up to about 2 billion, Euler Quartic Conjecture (in Russian).

Roger E. Frye, Finding 95800^4 + 217519^4 + 414560^4 = 422481^4 on the Connection Machine, Proceedings of Supercomputing 88, Vol. II: Science and Applications (1988), pp. 106-116.

Index to sequences related to Diophantine equations (4,1,3)

Example

The smallest solutions to a^4 + b^4 + c^4 = k^4 are (a,b,c,k) =
95800 217519 414560 422481 (Roger Frye)
673865 1390400 2767624 2813001 (Allan MacLeod)
1705575 5507880 8332208 8707481 (D. J. Bernstein)
5870000 8282543 11289040 12197457 (D. J. Bernstein)
4479031 12552200 14173720 16003017 (D. J. Bernstein)
3642840 7028600 16281009 16430513 (D. J. Bernstein)
2682440 15365639 18796760 20615673 (Noam Elkies)
2164632 31669120 41084175 44310257 (Robert Gerbicz)
10409096 42878560 65932985 68711097 (Robert Gerbicz)
34918520 87865617 106161120 117112081 (Robert Gerbicz)
1841160 121952168 122055375 145087793 (Juergen Rathmann)
27450160 108644015 146627384 156646737 (Juergen Rathmann)
186668000 260052385 582665296 589845921 (Seiji Tomita)
219076465 275156240 630662624 638523249 (Allan MacLeod)
558424440 606710871 769321280 873822121 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
588903336 859396455 1166705840 1259768473 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
50237800 632671960 1670617271 1679142729 (Seiji Tomita)
686398000 1237796960 1662997663 1787882337 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
92622401 1553556440 1593513080 1871713857 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
[Example lines revised by Robert Gerbicz, May 17 2009]

Crossrefs

See A078518, A078519, A078520 for values of a, b, c.

Cf. A134341, A175610.

Sequence in context: A264764 A130022 A175610 * A233722 A253958 A253965

Adjacent sequences: A003825 A003826 A003827 * A003829 A003830 A003831

Keyword

nonn,hard,nice

Author

N. J. A. Sloane

Extensions

More terms from Robert Gerbicz, Nov 13 2006

Extended by Robert Gerbicz, May 17 2009

Status

approved