A075760 Nontrivial binomial coefficients which are perfect powers (A001597).

36, 1225, 19600, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600

Offset: 1

Author

Robert G. Wilson v, Oct 08 2002

Keywords

• nonn
• more

Comments

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.

References

• 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.

Crossrefs

Cf. A001110.

Programs

Mathematica

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}]

Extensions

a(10)-a(12) from Sean A. Irvine, Mar 05 2025