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%I #46 Apr 03 2022 09:56:40
%S 0,1,3,8,120
%N Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a square.
%C 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
%C 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
%C 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
%C 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.
%C 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
%C 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
%D M. Gardner, "Mathematical Magic Show", M. Gardner, Alfred Knopf, New York, 1977, pp. 210, 221-222.
%D Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001, pp. 93-94.
%H A. Baker and H. Davenport, <a href="http://dx.doi.org/10.1093/qmath/20.1.129">The Equations 3x^2-2=y^2 and 8x^2-7=z^2</a>, Quart. J. Math. Oxford 20 (1969).
%H Nicolae Ciprian Bonciocat, Mihai Cipu, and Maurice Mignotte, <a href="https://arxiv.org/abs/2010.09200">There is no Diophantine D(-1)--quadruple</a>, arXiv:2010.09200 [math.NT], 2020.
%H Andrej Dujella, <a href="https://web.math.pmf.unizg.hr/~duje/dtuples.html">Diophantine m-tuples</a>
%H Z. Franusic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Franusic/franusic4.html">On the Extension of the Diophantine Pair {1,3} in Z[surd d]</a>, J. Int. Seq. 13 (2010) # 10.9.6
%H Zrinka Franušić, <a href="http://atlas-conferences.com/cgi-bin/abstract/cbbv-23">On the extension of the Diophantine pair {1, 3} in Z[√d]</a>, Journées Arithmétiques 2011. [Dead link]
%H Yasutsugu Fujita, <a href="http://dx.doi.org/10.1016/j.jnt.2009.01.001">Any Diophantine quintuple contains a regular Diophantine quadruple</a>, Journal of Number Theory, Volume 129, Issue 7, July 2009, Pages 1678-1697.
%H Martin Gardner, Mathematical diversions, Scientific American 216 (1967), <a href="https://www.jstor.org/stable/24931439">March 1967</a>, p. 124; <a href="https://www.jstor.org/stable/24931474">April 1967</a>, p. 119.
%Y Cf. A192629, A192630, A192631, A192632.
%K nonn,fini,full,nice
%O 0,3
%A Graham Lewis (grahaml(AT)levygee.com.uk)
%E Definition clarified by _Jonathan Sondow_, Jul 06 2011
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