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Search: id:A075760
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| A075760 |
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Nontrivial binomial coefficients which are perfect powers (A001597). |
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+0
1
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| 36, 1225, 19600, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Triangular-square numbers (A001110) are a subset, except for 0 and 1.
"For C(n,k) k>=4 and any l>=2 no solutions exist and this is what Erdos proved by an ingenious argument. ... C(50, 3) = 140^2 is the only solution for k = 3, l=2." page 13 of Aigner and Ziegler.
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REFERENCES
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Martin Aigner and Gunter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin, 2000, Chapter 3, "Binomial coefficients are (almost) never powers," pages 13-16.
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MATHEMATICA
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f[n_] := Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]]; a = {}; Do[ If[ f[n(n - 1)/2] > 1, a = Append[a, Binomial[n, 2]]]; If[ f[n(n - 1)*(n - 2)/6] > 1, a = Append[a, Binomial[n, 3]]], {n, 5, 1500000}]
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CROSSREFS
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Cf. A001110.
Sequence in context: A151584 A103278 A004294 this_sequence A113938 A001110 A064196
Adjacent sequences: A075757 A075758 A075759 this_sequence A075761 A075762 A075763
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KEYWORD
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nonn
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AUTHOR
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Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 08 2002
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