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.

6, 71, 135, 372, 426, 242, 566, 791, 236, 575, 1938, 2676, 1124, 2196, 1943, 1851, 1943, 7676, 3318, 10866, 3086, 3453, 17328, 4607, 28182, 10230, 25765, 31212, 7251, 34199, 6560, 15218, 29196, 54101, 32882, 51293, 17384, 8999, 58462, 75263

Offset

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)

References

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.

Links

Jean-François Alcover, Table of n, a(n) for n = 1..60

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

Example

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

Crossrefs

Cf. A050787, A050789, A050790.

Sequence in context: A182563 A211036 A284215 * A241883 A297746 A027317

Adjacent sequences: A050785 A050786 A050787 * A050789 A050790 A050791

Keyword

nonn

Author

Patrick De Geest, Sep 15 1999

Extensions

More terms from Jud McCranie, Dec 25 2000

Further terms from Don Reble, Nov 29 2001

Status

approved