A192629 -id:A192629 - 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

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.

A192631 Numerators of the Diophantus-Dujella rational Diophantine quintuple: 1 + the product of any two distinct terms is a square.

Original entry on oeis.org

1, 33, 17, 105, 549120

Offset: 1

Jonathan Sondow, Jul 07 2011

Denominators are A192632. Diophantus found the rational Diophantine quadruple 1/16, 33/16, 17/4, 105/16. Dujella added a fifth rational number 549120/10201.
It is unknown whether this rational Diophantine quintuple can be extended to a sextuple. Herrmann, Pethoe, and Zimmer proved that the sequence is finite, but no bound on its length is known.
See A030063 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.

E. Herrmann, A. Pethoe and H. G. Zimmer, On Fermat's quadruple equations, Abh. Math. Sem. Univ. Hamburg 69 (1999), 283-291.

A. Dujella, Rational Diophantine m-tuples

Cf. A030063, A192629, A192630, 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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A192630 Denominators of the Fermat-Euler rational Diophantine m-tuple.

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A192631 Numerators of the Diophantus-Dujella rational Diophantine quintuple: 1 + the product of any two distinct terms is a square.

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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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