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A030063
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Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a square.
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5
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OFFSET
| 0,3
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COMMENTS
| 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
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. [From Alonso del Arte, (alonso.delarte(AT)gmail.com), 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.
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REFERENCES
| A. Baker and H. Davenport, The Equations 3x^2-2=y^2 and 8x^2-7=z^2, Quart. J. Math. Oxford 20 (1969).
M. Gardner, "Mathematical Magic Show", M. Gardner, Alfred Knopf, New York, 1977, pp. 210, 221-222.
M. Gardner, Mathematical diversions, Scientific American 216 (1967), March 1967, p. 124; April 1967, p. 119.
Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001, pp. 93-94.
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LINKS
| Andrej Dujella (duje(AT)math.hr), Diophantine m-tuples
Dujella A.,Diophantine m-tuple, [From Artur Jasinski (grafix(AT)csl.pl), Feb 11 2010]
Zrinka Franušić,On the extension of the Diophantine pair {1, 3} in Z[√d], Journées Arithmétiques 2011.
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CROSSREFS
| Cf. A192629, A192630, A192631, A192632.
Sequence in context: A028504 A123279 A134803 * A195568 A051047 A192629
Adjacent sequences: A030060 A030061 A030062 * A030064 A030065 A030066
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KEYWORD
| nonn,fini,full,nice
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AUTHOR
| Graham Lewis (grahaml(AT)levygee.com.uk)
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EXTENSIONS
| Definition clarified by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 06 2011
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