A050792 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

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

EXAMPLE

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

CROSSREFS

Cf. A050791, A050793, A050794.

Sequence in context: A342197 A302975 A165447 * A171671 A016886 A099761