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

7, 10, 24, 26, 65, 13777

Offset: 1

Author

Ctibor O. Zizka, May 29 2010

Keywords

• nonn
• more
• fini

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 10 million. - Harvey P. Dale, Apr 04 2017

No additional terms up to 10 billion. - Jon E. Schoenfield, Mar 18 2022

No additional terms up to 1 trillion. The sequence is finite by Siegel's theorem on integral points. - David Radcliffe, Jan 01 2024

Links

• Wikipedia, Tetrahedral number

Crossrefs

Cf. A006451, A007318, A055997, A072221.

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

Programs

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 *)

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):
        print(m) # Mohammed Yaseen, Mar 18 2022