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

1,2

Comments

This sequence probably contains no more than 5 primes.

Links

Table of n, a(n) for n=1..12.

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]

Formula

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.

Example

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

Prog

(PARI) { 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 */

Crossrefs

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

Sequence in context: A089971 A082620 A199304 * A103992 A185949 A001387

Adjacent sequences: A156665 A156666 A156667 * A156669 A156670 A156671

Keyword

nonn

Author

Max Alekseyev, Feb 13 2009

Status

approved