A003828 - OEIS

%I #38 Sep 24 2023 04:17:41

%S 422481,2813001,8707481,12197457,16003017,16430513,20615673,44310257,

%T 68711097,117112081,145087793,156646737,589845921,638523249,873822121,

%U 1259768473,1679142729,1787882337,1871713857

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

%C There are no further terms up to 1986560000. - _Robert Gerbicz_, May 17 2009

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

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

%H N. Elkies, <a href="https://doi.org/10.1090/S0025-5718-1988-0930224-9">On A^4 + B^4 + C^4 = D^4</a>, Math. Comp. 51:184 (1988), 825-835.

%H D. J. Bernstein, <a href="https://doi.org/10.1090/S0025-5718-00-01219-9">Enumerating solutions to p(a) + q(b) = r(c) + s(d)</a>, Math. Comp. 70:233 (2001), 389-394.

%H Finished distributed computation for solutions up to about 2 billion, <a href="http://euler413.narod.ru/">Euler Quartic Conjecture</a> (in Russian).

%H Roger E. Frye, <a href="https://doi.org/10.1109%2FSUPERC.1988.74138">Finding 95800^4 + 217519^4 + 414560^4 = 422481^4 on the Connection Machine</a>, Proceedings of Supercomputing 88, Vol. II: Science and Applications (1988), pp. 106-116.

%H <a href="/index/Di#Diophantine">Index to sequences related to Diophantine equations</a> (4,1,3)

%e The smallest solutions to a^4 + b^4 + c^4 = k^4 are (a,b,c,k) =

%e 95800 217519 414560 422481 (Roger Frye)

%e 673865 1390400 2767624 2813001 (Allan MacLeod)

%e 1705575 5507880 8332208 8707481 (D. J. Bernstein)

%e 5870000 8282543 11289040 12197457 (D. J. Bernstein)

%e 4479031 12552200 14173720 16003017 (D. J. Bernstein)

%e 3642840 7028600 16281009 16430513 (D. J. Bernstein)

%e 2682440 15365639 18796760 20615673 (Noam Elkies)

%e 2164632 31669120 41084175 44310257 (Robert Gerbicz)

%e 10409096 42878560 65932985 68711097 (Robert Gerbicz)

%e 34918520 87865617 106161120 117112081 (Robert Gerbicz)

%e 1841160 121952168 122055375 145087793 (Juergen Rathmann)

%e 27450160 108644015 146627384 156646737 (Juergen Rathmann)

%e 186668000 260052385 582665296 589845921 (Seiji Tomita)

%e 219076465 275156240 630662624 638523249 (Allan MacLeod)

%e 558424440 606710871 769321280 873822121 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)

%e 588903336 859396455 1166705840 1259768473 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)

%e 50237800 632671960 1670617271 1679142729 (Seiji Tomita)

%e 686398000 1237796960 1662997663 1787882337 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)

%e 92622401 1553556440 1593513080 1871713857 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)

%e [Example lines revised by _Robert Gerbicz_, May 17 2009]

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

%Y Cf. A134341, A175610.

%K nonn,hard,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Robert Gerbicz_, Nov 13 2006

%E Extended by _Robert Gerbicz_, May 17 2009