A339326 Denominators of s in rational solutions of s^4 + s^3 + s^2 + s + 1 = y^2 with |s| <= 1.

1, 1, 3, 11, 123, 808, 43993, 1404304, 113095273, 16258517264, 5907678749271, 1749162037068984, 2230703155726839733, 2430407134728632414424, 9811722627654286580946253, 28104484948123389151863529007, 447820184835469405718954028342863, 5093605667758828993168776807306887631

Offset: 1

Author

Jeremy Tan, Nov 30 2020

Keywords

• nonn
• frac

Comments

If abs(A339325(n)) = 1 or a(n) = 1 then n <= 3, i.e., the only integer solutions to s^4 + s^3 + s^2 + s + 1 = y^2 are (s, y) = (-1, +-1), (0, +-1), (3, +-11). This may easily be shown by bounding the LHS between two consecutive perfect squares.

Examples

			The values of s in solutions (s, y) with |s| <= 1 begin -1, 0, 1/3, -8/11, -35/123, 627/808, -20965/43993, ...
		

Links

• Jeremy Tan, Table of n, a(n) for n = 1..50
• Jeremy Tan, Rigid pentagons and rational solutions of s^4+s^3+s^2+s+1=y^2, Mathematics Stack Exchange, Apr 1 2020.
• Gerard 't Hooft, Meccano Math I

Crossrefs

Cf. A339325 (numerators).

Programs

Mathematica

a[1] = 1; a[2] = 1; a[n_] := Module[{x = 4, y = 2, s, xr}, Do[s = (y-1) / (x-1); xr = s^2 - x + 4; {x, y} = {xr, s(x-xr) - y}, n-2]; s = (2y-x) / (4x-5); Denominator[MinimalBy[{s, 1/s}, Abs][[1]]]]; Table[a[k], {k, 20}] (* Jeremy Tan, Nov 15 2021 *)

PARI

a(n) = {
    [u,v] = ellmul(ellinit([0,-5,0,5,0]), [1,1], n);
    s = (2*v-u) / (4*u-5);
    if(abs(s)>1, s=1/s);
    denominator(s)
}