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 A050788 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. 5

%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. 107-124.

%D David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.

%H Jean-François 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

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