|
Search: id:A050792
|
|
|
| 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. |
|
+0
5
|
|
| 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; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
"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.
David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.
|
|
LINKS
|
Lewis Mammel, Table of n, a(n) for n = 1..368
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers
|
|
EXAMPLE
|
E.g. (577)^3 + 2304^3 = 2316^3 + 1.
|
|
CROSSREFS
|
Cf. A050791, A050793, A050794.
Sequence in context: A165510 A165749 A165447 this_sequence A016886 A099761 A092396
Adjacent sequences: A050789 A050790 A050791 this_sequence A050793 A050794 A050795
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Patrick De Geest (pdg(AT)worldofnumbers.com), Sep 15 1999.
|
|
EXTENSIONS
|
More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) No more with z<8192.
Extended through 26914 by Jud McCranie (j.mccranie(AT)comcast.net), Dec 25 2000
More terms from Don Reble (djr(AT)nk.ca), Nov 29 2001
Edited by N. J. A. Sloane (njas(AT)research.att.com), May 08 2007
|
|
|
Search completed in 0.002 seconds
|
