1,1

Comments from Fred W. Helenius (fredh(AT)ix.netcom.com), Jul 22 2008: (Start)

There is an infinite family of solutions to c^3+1=a^3+b^3 given by

(a,b,c) = (9n^3 + 1, 9n^4, 9n^4 + 3n). The present sequence actually asks about

x^3+y^3=z^3-1 with x < y < z; for that we can take

(x,y,z) = (9n^3 - 1, 9n^4 - 3n, 9n^4) for n > 1.

I extracted these solutions from Theorem 235 in Hardy & Wright; the

result shown there is that all nontrivial rational solutions of

x^3 + y^3 = u^3 + v^3 are given by

x = r(1 - (a - 3b)(a^2 + 3b^2))

y = r((a + 3b)(a^2 + 3b^2) - 1)

u = r((a + 3b) - (a^2 + 3b^2)^2)

v = r((a^2 + 3b^2)^2 - (a - 3b))

where r,a,b are rational and r is not zero.

Specializing to r = 1, b = n/2 and a = 3n/2 gives

x = 1, y = 9n^3 - 1, u = 3n - 9n^4, v = 9n^4.

The solutions given above are obtained by changing signs and moving

cubes from one side of the equation to the other as necessary.

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)

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

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

Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

E.g. (575)^3 + 2292^3 = 2304^3 - 1.

Cf. A050787, A050789, A050790.

Sequence in context: A001448 A024489 A036361 this_sequence A027317 A099339 A023038

Adjacent sequences: A050785 A050786 A050787 this_sequence A050789 A050790 A050791

nonn

Patrick De Geest (pdg(AT)worldofnumbers.com), Sep 15 1999.

More terms from Jud McCranie (j.mccranie(AT)comcast.net), Dec 25 2000

Further terms from Don Reble (djr(AT)nk.ca), Nov 29 2001