%I #24 Oct 17 2019 14:26:04
%S 6,71,135,372,426,242,566,791,236,575,1938,2676,1124,2196,1943,1851,
%T 1943,7676,3318,10866,3086,3453,17328,4607,28182,10230,25765,31212,
%U 7251,34199,6560,15218,29196,54101,32882,51293,17384,8999,58462,75263
%N Consider the Diophantine equation x^3 + y^3 = z^3  1 (x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of x.
%C From Fred W. Helenius (fredh(AT)ix.netcom.com), Jul 22 2008: (Start)
%C There is an infinite family of solutions to c^3+1=a^3+b^3 given by
%C (a,b,c) = (9n^3 + 1, 9n^4, 9n^4 + 3n). The present sequence actually asks about
%C x^3 + y^3 = z^3  1 with x < y < z; for that we can take
%C (x,y,z) = (9n^3  1, 9n^4  3n, 9n^4) for n > 1.
%C I extracted these solutions from Theorem 235 in Hardy & Wright; the result shown there is that all nontrivial rational solutions of
%C x^3 + y^3 = u^3 + v^3 are given by
%C x = r(1  (a  3b)(a^2 + 3b^2))
%C y = r((a + 3b)(a^2 + 3b^2)  1)
%C u = r((a + 3b)  (a^2 + 3b^2)^2)
%C v = r((a^2 + 3b^2)^2  (a  3b))
%C where r,a,b are rational and r is not zero.
%C Specializing to r = 1, b = n/2 and a = 3n/2 gives
%C x = 1, y = 9n^3  1, u = 3n  9n^4, v = 9n^4.
%C The solutions given above are obtained by changing signs and moving cubes from one side of the equation to the other as necessary.
%C Unfortunately, not all integral solutions are found so easily: the third value in A050788 corresponds to 135^3 + 138^3 = 172^3  1; this is not produced by such simple choices of r,a,b. (End)
%D Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107124.
%D David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.
%H JeanFrançois Alcover, <a href="/A050788/b050788.txt">Table of n, a(n) for n = 1..60</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">Diophantine Equation  3rd Powers</a>
%e (575)^3 + 2292^3 = 2304^3  1.
%Y Cf. A050787, A050789, A050790.
%K nonn
%O 1,1
%A _Patrick De Geest_, Sep 15 1999
%E More terms from _Jud McCranie_, Dec 25 2000
%E Further terms from _Don Reble_, Nov 29 2001
