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
Keywords
Examples
13361 belongs to this sequence since 13361^2 = (35^5 + 123^5) / (35 + 123) with gcd(35, 123)=1.
Links
- 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]
Programs
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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 */
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.
Comments