A192629 - OEIS

%I #26 Jun 01 2024 23:51:44

%S 0,1,3,8,120,777480

%N Numerators of the Fermat-Euler rational Diophantine m-tuple.

%C 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.

%C 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.

%C 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

%C Denominators are A192630.

%C See A030063 for additional comments, references, and links.

%H A. Dujella, <a href="http://web.math.hr/~duje/ratio.html">Rational Diophantine m-tuples</a>

%H E. Herrmann, A. Pethoe and H. G. Zimmer, <a href="https://doi.org/10.1007/BF02940880">On Fermat's quadruple equations</a>, Abh. Math. Sem. Univ. Hamburg 69 (1999), 283-291.

%H Michael Stoll, <a href="https://doi.org/10.4064/aa180416-4-10">Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples</a>, Acta Arith. 190 (2019), 239-261.

%e 0/1, 1/1, 3/1, 8/1, 120/1, 777480/8288641.

%e 1 + 1*(777480/8288641) = (3011/2879)^2.

%Y Cf. A030063, A192630, A192631, A192632.

%K nonn,fini,full,frac

%O 0,3

%A _Jonathan Sondow_, Jul 06 2011