A339326 -id:A339326 - cp's OEIS frontend

A156668 Positive integers k such that k^2 = (m^5 + n^5)/(m + n) for some coprime integers m, n.

1, 11, 101, 13361, 1169341, 1612186411, 1624763543401, 20188985439712961, 240020196429554642201, 29891946989942513908518251, 3506790234728288196345900732301, 5190947078637547438603476743093680561

Offset: 1

Max Alekseyev, Feb 13 2009

nonn

This sequence probably contains no more than 5 primes.

			13361 belongs to this sequence since 13361^2 = (35^5 + 123^5) / (35 + 123) with gcd(35, 123)=1.

Kevin Acres and David Broadhurst, Rational points on y^2 = x^3 + 10*x^2 + 5*x
Dave Rusin, Re: Help with diophantine equ., sci.math newsgroup [Broken link]
Dave Rusin, Re: Help with diophantine equ., sci.math newsgroup [Cached copy]

Cf. A156669, A156670, A186082, A339325 (m), A339326 (n).

{ a(k) = local(P=ellpow(ellinit([0,10,0,5,0]),[-1,2],k),s,t); s=P[1]^2;t=abs(numerator(P[2]^4/s-80*s)); while(t%2==0,t=t/2); t } /* David Broadhurst */

Numerators of rational numbers (81*x^4 + 540*x^3 - 8370*x^2 + 33900*x - 47975)/(9*x^2 - 150*x + 445)^2, where x ranges over abscissas of rational points on the elliptic curve y^2 = x^3 - 85/3*x + 1550/27.

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

Original entry on oeis.org

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

Offset: 1

Jeremy Tan, Nov 30 2020

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.

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

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

Cf. A339326 (denominators).

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 *)

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)
}

cp's OEIS Frontend

A156668 Positive integers k such that k^2 = (m^5 + n^5)/(m + n) for some coprime integers m, n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula