A192631 -id:A192631 - cp's OEIS frontend

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

0, 1, 3, 8, 120

Offset: 0

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

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

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.

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.

Cf. A192629, A192630, A192631, A192632.

Definition clarified by Jonathan Sondow, Jul 06 2011

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

Original entry on oeis.org

0, 1, 3, 8, 120, 777480

Offset: 0

Jonathan Sondow, Jul 06 2011

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.

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

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.

Cf. A030063, A192630, A192631, A192632.

A192630 Denominators of the Fermat-Euler rational Diophantine m-tuple.

Original entry on oeis.org

1, 1, 1, 1, 1, 8288641

Offset: 0

Jonathan Sondow, Jul 06 2011

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.
Stoll proved that an extension of Fermat's set to a rational quintuple with the same property is unique. - Andrej Dujella, May 12 2024
Numerators are A192629.
See A030063 and A192629 for additional comments, references, and links.

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

Michael Stoll, Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples, Acta Arith. 190 (2019), 239-261.

Cf. A030063, A192629, A192631, A192632.

A192632 Denominators of the Diophantus-Dujella rational Diophantine quintuple: 1 + the product of any two distinct terms is a square.

Original entry on oeis.org

16, 16, 4, 16, 10201

Offset: 1

Jonathan Sondow, Jul 07 2011

Numerators are A192631. Diophantus found the rational Diophantine quadruple 1/16, 33/16, 17/4, 105/16. Dujella added a fifth rational number 549120/10201.

See A030063 and A192631 for additional comments, references, and links.

			1/16, 33/16, 17/4, 105/16, 549120/10201.
1 + (1/16)*(33/16) = (17/16)^2.
1 + (33/16)*(549120/10201) = (1069/101)^2.

A. Dujella, Rational Diophantine m-tuples

Cf. A030063, A192629, A192630, A192631.

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

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

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A192630 Denominators of the Fermat-Euler rational Diophantine m-tuple.

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A192632 Denominators of the Diophantus-Dujella rational Diophantine quintuple: 1 + the product of any two distinct terms is a square.

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