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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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9
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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Martin Gardner, Mathematical diversions, Scientific American 216 (1967), March 1967, p. 124; April 1967, p. 119.
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CROSSREFS
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KEYWORD
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nonn,fini,full,nice
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AUTHOR
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Graham Lewis (grahaml(AT)levygee.com.uk)
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EXTENSIONS
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STATUS
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approved
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