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

7, 10, 24, 26, 65, 13777

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 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

Table of n, a(n) for n=1..6.

Wikipedia, Tetrahedral number

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

Crossrefs

Cf. A006451, A007318, A055997, A072221.

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

Sequence in context: A183330 A196316 A134329 * A241052 A280173 A229310

Adjacent sequences: A175489 A175490 A175491 * A175493 A175494 A175495

Keyword

nonn,more,fini

Author

Ctibor O. Zizka, May 29 2010

Status

approved