

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

Martin Gardner, Mathematical diversions, Scientific American 216 (1967), March 1967, p. 124; April 1967, p. 119.


CROSSREFS



KEYWORD

nonn,fini,full,nice


AUTHOR

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


EXTENSIONS



STATUS

approved



