A192629 Numerators of the Fermat-Euler rational Diophantine m-tuple.

0, 1, 3, 8, 120, 777480

Offset: 0

Author

Jonathan Sondow, Jul 06 2011

Keywords

• nonn
• fini
• full
• frac

Comments

Fermat gave the integer Diophantine m-tuple 1, 3, 8, 120 (see A030063): 1 + the product of any two distinct terms is a square. Euler added the rational number 777480/8288641.

It was unknown whether this rational Diophantine m-tuple can be extended by another rational number. Herrmann, Pethoe, and Zimmer proved that the sequence is finite, but no bound on its length is known.

In 2019, Stoll proved that an extension of Fermat's set to a rational quintuple with the same property is unique. Thus, the quintuple 1, 3, 8, 120, 777480/8288641 cannot be extended to a rational Diophantine sextuple. - Andrej Dujella, May 12 2024

Denominators are A192630.

See A030063 for additional comments, references, and links.

Examples

			0/1, 1/1, 3/1, 8/1, 120/1, 777480/8288641.
1 + 1*(777480/8288641) = (3011/2879)^2.
		

Links

• A. Dujella, Rational Diophantine m-tuples
• E. Herrmann, A. Pethoe and H. G. Zimmer, On Fermat's quadruple equations, Abh. Math. Sem. Univ. Hamburg 69 (1999), 283-291.
• Michael Stoll, Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples, Acta Arith. 190 (2019), 239-261.

Crossrefs

Cf. A030063, A192630, A192631, A192632.