A030063 Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a square.

0, 1, 3, 8, 120

Offset: 0

Author

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

Keywords

• nonn
• fini
• full
• nice

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, 221-222.
• Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001, pp. 93-94.

Links

• A. Baker and H. Davenport, The Equations 3x^2-2=y^2 and 8x^2-7=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 m-tuples
• 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 1678-1697.
• Martin Gardner, Mathematical diversions, Scientific American 216 (1967), March 1967, p. 124; April 1967, p. 119.

Crossrefs

Cf. A192629, A192630, A192631, A192632.

Extensions

Definition clarified by Jonathan Sondow, Jul 06 2011