A050792 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z (see A050791). Sequence gives values of x.

9, 64, 73, 135, 334, 244, 368, 1033, 1010, 577, 3097, 3753, 1126, 4083, 5856, 3987, 1945, 11161, 13294, 3088, 10876, 16617, 4609, 27238, 5700, 27784, 11767, 26914, 38305, 6562, 49193, 27835, 35131, 7364, 65601, 50313, 9001, 11980, 39892, 20848

Offset: 1

Author

Patrick De Geest, Sep 15 1999

Keywords

• nonn

Comments

"One of the simplest cubic Diophantine equations is known to have an infinite number of solutions (Lehmer, 1956; Payne and Vaserstein, 1991). Any number of solutions to the equation x^3 + y^3 + z^3 = 1 can be produced through the use of the algebraic identity (9t^3+1)^3 + (9t^4)^3 + (-9t^4-3t)^3 = 1 by substituting in values of t. ...

"Although these are certainly solutions, the identity generates only one family of solutions. Other solutions such as (94, 64, -103), (235, 135, -249), (438, 334, -495), ... can be found. What is not known is if it is possible to parameterize all solutions for this equation. Put another way, are there an infinite number of families of solutions? Probable yes, but that too remains to be shown." [Herkommer]

Values of x associated with A050794.

Examples

			577^3 + 2304^3 = 2316^3 + 1.
		

References

• Mark A. Herkommer, Number Theory, A Programmer's Guide, McGraw-Hill, NY, 1999, page 370.
• Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Links

• Lewis Mammel, Table of n, a(n) for n = 1..368
• Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Crossrefs

Cf. A050791, A050793, A050794.

Extensions

More terms from Michel ten Voorde.

Extended through 26914 by Jud McCranie, Dec 25 2000

More terms from Don Reble, Nov 29 2001

Edited by N. J. A. Sloane, May 08 2007