

A175492


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


1




OFFSET

1,1


COMMENTS

Related sequences:
Numbers m such that binomial(m,2) is a square: A055997;
Numbers m such that binomial(m,2) + 1 is a square: A006451 + 1;
Numbers m such that binomial(m,2)  1 is a square: A072221 + 1;
Numbers m >= 3 such that binomial(m,3) is a square: {3, 4, 50} (Proved by A. J. Meyl in 1878);
Numbers m >= 4 such that binomial(m,4) + 1 is a square: {6, 7, 45, 55, ...};
Numbers m >= 7 such that binomial(m,7) + 1 is a square: {8, 10, 21, 143, ...}.
No additional terms up to 1 trillion. The sequence is finite by Siegel's theorem on integral points.  David Radcliffe, Jan 01 2024


LINKS



MATHEMATICA

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 *)
Select[Range[3, 14000], IntegerQ[Sqrt[Binomial[#, 3]+1]]&] (* Harvey P. Dale, Apr 04 2017 *)


PROG

(PARI) isok(m) = (m>=3) && issquare(binomial(m, 3)+1); \\ Michel Marcus, Mar 15 2022
(Python)
from sympy import binomial
from sympy.ntheory.primetest import is_square
for m in range(3, 10**6):
if is_square(binomial(m, 3)+1):


CROSSREFS

Cf. A216268 (values of binomial(m, 3)) and A216269 (square roots of binomial(m, 3) + 1).


KEYWORD

nonn,more,fini


AUTHOR



STATUS

approved



