A339325 Numerators of s in rational solutions of s^4 + s^3 + s^2 + s + 1 = y^2 with |s| <= 1.

-1, 0, 1, -8, -35, 627, -20965, -761577, 72676071, -3470319335, -4692803610731, 418741237461085, 180890439981934931, -2244276655546627749157, -764583000726654718413105, 12199909914654265034887926688, -296226554714255082163286916350895, -1802246724473363548181037369907741088

Offset: 1

Author

Jeremy Tan, Nov 30 2020

Keywords

• sign
• frac

Comments

If (s, y) is a solution then so is (s, -y); if s != 0 then (1/s, y/s^2) is also a solution.

The quartic elliptic curve is birationally equivalent to v^2 = u^3 - 5*u^2 + 5*u with s = (2*v-u) / (4*u-5). All solutions can be generated from multiples of (u,v) = P = (1,1) and the two transformations above.

Let (s, y) be a solution, a = 1 + s, b = 1 + 1/s and c = |y/s|. Then the distance between a*exp(3*i*Pi/5) and 1 + b*exp(2*i*Pi/5) is c, with a, b, c all rational. This allows creating a rigid regular pentagon with idealized Meccano strips - see 't Hooft for the solution corresponding to 3P, and the Mathematics Stack Exchange link for the derivation and solution corresponding to 4P.

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. A339326 (denominators).

Programs

Mathematica

a[1] = -1; a[2] = 0; 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); Numerator[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);
    numerator(s)
}