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

1,4

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.

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

Example

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

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 *)

Prog

(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)
}

Crossrefs

Cf. A339326 (denominators).

Sequence in context: A000426 A250889 A200312 * A089698 A133887 A200713

Adjacent sequences: A339322 A339323 A339324 * A339326 A339327 A339328

Keyword

sign,frac

Author

Jeremy Tan, Nov 30 2020

Status

approved