A050792 - OEIS

%I #31 Nov 30 2019 09:05:10

%S 9,64,73,135,334,244,368,1033,1010,577,3097,3753,1126,4083,5856,3987,

%T 1945,11161,13294,3088,10876,16617,4609,27238,5700,27784,11767,26914,

%U 38305,6562,49193,27835,35131,7364,65601,50313,9001,11980,39892,20848

%N 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.

%C "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. ...

%C "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]

%C Values of x associated with A050794.

%D Mark A. Herkommer, Number Theory, A Programmer's Guide, McGraw-Hill, NY, 1999, page 370.

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

%H Lewis Mammel, <a href="/A050792/b050792.txt">Table of n, a(n) for n = 1..368</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">Diophantine Equation - 3rd Powers</a>

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

%Y Cf. A050791, A050793, A050794.

%K nonn

%O 1,1

%A _Patrick De Geest_, Sep 15 1999

%E More terms from _Michel ten Voorde_.

%E Extended through 26914 by _Jud McCranie_, Dec 25 2000

%E More terms from _Don Reble_, Nov 29 2001

%E Edited by _N. J. A. Sloane_, May 08 2007