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Nontrivial binomial coefficients which are perfect powers (A001597).
0

%I #4 Mar 30 2012 17:30:50

%S 36,1225,19600,41616,1413721,48024900,1631432881,55420693056,

%T 1882672131025

%N Nontrivial binomial coefficients which are perfect powers (A001597).

%C Triangular-square numbers (A001110) are a subset, except for 0 and 1.

%C "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.

%D 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.

%t 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}]

%Y Cf. A001110.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Oct 08 2002