A341671 Solutions y of the Diophantine equation 3*(x^2+x+1) = y^2.

3, 39, 543, 7563, 105339, 1467183, 20435223, 284625939, 3964327923, 55215964983, 769059181839, 10711612580763, 149193516948843, 2077997624703039, 28942773228893703, 403120827579808803, 5614748812888429539, 78203362552858204743, 1089232326927126436863, 15171049214426911911339

Offset: 1

Author

Bernard Schott, Feb 17 2021

Keywords

• nonn

Comments

Corresponding x are in A028231.

This equation belongs to the family of equations studied by Kustaa A. Inkeri, y^m = a * (x^q-1)/(x-1) with here: m=2, a=3, q=3. This equation is exhibed in A307745 by Giovanni Resta to prove that this sequence has infinitely many terms.

This Diophantine equation 3*(x^2+x+1) = y^2 has infinitely many solutions because the Pell-Fermat equation u^2 - 3*v^2 = -2 also has infinitely many solutions. The corresponding (u,v) are in (A001834, A001835) and for each pair (u,v), the corresponding solutions of 3*(x^2+x+1) = y^2 are x = (3*u*v-1)/2 and y = 3*(u^2+1)/2.

Note that if y = 3*z, this equation becomes 3*z^2 = x^2+x+1 with solutions (x, z) = (A028231, A001570).

Examples

			The first few values for (x,y) are (1,3), (22,39), (313,543), (4366,7563), (60817,105339), ...
		

Links

• Kustaa A. Inkeri, On the Diophantine equation a(x^n-1)/(x-1) = y^m, Acta Arithmetica, Vol. 21, No. 1 (1972), pp. 299-311.

Crossrefs

Cf. A001834, A001835, A001570, A028231, A307745.

Subsequence of A158235, for a(n)>3.

Programs

Mathematica

f[x_] := Sqrt[3*(x^2 + x + 1)]; f /@ LinearRecurrence[{15, -15, 1}, {1, 22, 313}, 20] (* Amiram Eldar, Feb 17 2021 *)

Formula

a(n) = 3*A001570(n). - Hugo Pfoertner, Feb 17 2021

a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).

Extensions

More terms from Amiram Eldar, Feb 17 2021