

A030063


Fermat's Diophantine mtuple: 1 + the product of any two distinct terms is a square.


9




OFFSET

0,3


COMMENTS

Baker and Davenport proved that no other positive integer can replace 120 and still preserve the property that 1 + the product of any two distinct terms is a square. In particular, the sequence cannot be extended to another integer term. However, it can be extended to another rational term  see A192629.  Jonathan Sondow, Jul 11 2011
It is conjectured that there do not exist five strictly positive integers with the property that 1 + the product of any two distinct terms is a square. (See Dujella's links.)  Jonathan Sondow, Apr 04 2013
Other such quadruples can be generated using the formula F(2n), F(2n + 2), F(2n + 4) and F(2n + 1)F(2n + 2)F(2n + 3) given in Koshy's book.  Alonso del Arte, Jan 18 2011
Other such quadruples are generated by Euler's formula a, b, a+b+2*r, 4*r*(r+a)*(r+b), where 1+a*b = r^2.
Seems to be equivalent to: 1 + the product of any two distinct terms is a perfect power. Tested up to 10^10.  Robert C. Lyons, Jun 30 2016
Seems to be equivalent to: 1 + the product of any two distinct terms is a powerful number. Tested up to 1.2*10^9.  Robert C. Lyons, Jun 30 2016


REFERENCES

M. Gardner, "Mathematical Magic Show", M. Gardner, Alfred Knopf, New York, 1977, pp. 210, 221222.
Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001, pp. 9394.


LINKS

Table of n, a(n) for n=0..4.
A. Baker and H. Davenport, The Equations 3x^22=y^2 and 8x^27=z^2, Quart. J. Math. Oxford 20 (1969).
Nicolae Ciprian Bonciocat, Mihai Cipu, and Maurice Mignotte, There is no Diophantine D(1)quadruple, arXiv:2010.09200 [math.NT], 2020.
Andrej Dujella, Diophantine mtuples
Z. Franusic, On the Extension of the Diophantine Pair {1,3} in Z[surd d], J. Int. Seq. 13 (2010) # 10.9.6
Zrinka Franušić, On the extension of the Diophantine pair {1, 3} in Z[√d], Journées Arithmétiques 2011. [Dead link]
Yasutsugu Fujita, Any Diophantine quintuple contains a regular Diophantine quadruple, Journal of Number Theory, Volume 129, Issue 7, July 2009, Pages 16781697.
Martin Gardner, Mathematical diversions, Scientific American 216 (1967), March 1967, p. 124; April 1967, p. 119.


CROSSREFS

Cf. A192629, A192630, A192631, A192632.
Sequence in context: A123279 A279164 A134803 * A195568 A051047 A192629
Adjacent sequences: A030060 A030061 A030062 * A030064 A030065 A030066


KEYWORD

nonn,fini,full,nice


AUTHOR

Graham Lewis (grahaml(AT)levygee.com.uk)


EXTENSIONS

Definition clarified by Jonathan Sondow, Jul 06 2011


STATUS

approved



