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A175492 Numbers m >= 3 such that binomial(m,3) + 1 is a square.

%I A175492 #37 Jan 02 2024 15:16:08
%S A175492 7,10,24,26,65,13777
%N A175492 Numbers m >= 3 such that binomial(m,3) + 1 is a square.
%C A175492 Related sequences:
%C A175492 Numbers m such that binomial(m,2) is a square: A055997;
%C A175492 Numbers m such that binomial(m,2) + 1 is a square: A006451 + 1;
%C A175492 Numbers m such that binomial(m,2) - 1 is a square: A072221 + 1;
%C A175492 Numbers m >= 3 such that binomial(m,3) is a square: {3, 4, 50} (Proved by A. J. Meyl in 1878);
%C A175492 Numbers m >= 4 such that binomial(m,4) + 1 is a square: {6, 7, 45, 55, ...};
%C A175492 Numbers m >= 7 such that binomial(m,7) + 1 is a square: {8, 10, 21, 143, ...}.
%C A175492 No additional terms up to 10 million. - _Harvey P. Dale_, Apr 04 2017
%C A175492 No additional terms up to 10 billion. - _Jon E. Schoenfield_, Mar 18 2022
%C A175492 No additional terms up to 1 trillion. The sequence is finite by Siegel's theorem on integral points. - _David Radcliffe_, Jan 01 2024
%H A175492 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetrahedral_number">Tetrahedral number</a>
%t A175492 lst = {}; k = 3; While[k < 10^6, If[ IntegerQ@ Sqrt[ Binomial[k, 3] + 1], AppendTo[lst, k]]; k++ ]; lst (* _Robert G. Wilson v_, Jun 11 2010 *)
%t A175492 Select[Range[3,14000],IntegerQ[Sqrt[Binomial[#,3]+1]]&] (* _Harvey P. Dale_, Apr 04 2017 *)
%o A175492 (PARI) isok(m) = (m>=3) && issquare(binomial(m,3)+1); \\ _Michel Marcus_, Mar 15 2022
%o A175492 (Python)
%o A175492 from sympy import binomial
%o A175492 from sympy.ntheory.primetest import is_square
%o A175492 for m in range(3, 10**6):
%o A175492     if is_square(binomial(m,3)+1):
%o A175492         print(m) # _Mohammed Yaseen_, Mar 18 2022
%Y A175492 Cf. A006451, A007318, A055997, A072221.
%Y A175492 Cf. A216268 (values of binomial(m, 3)) and A216269 (square roots of binomial(m, 3) + 1).
%K A175492 nonn,more,fini
%O A175492 1,1
%A A175492 _Ctibor O. Zizka_, May 29 2010