A117319 Values of n for which Leech's problem "Find two rational right-angled triangles on the same base whose heights are in the ratio n:1" has a solution.

7, 10, 11, 12, 14, 17, 19, 22, 23, 27, 28, 29, 30, 33, 38, 39, 40, 41, 42, 44, 45, 47, 48, 51, 52, 53, 54, 57, 58, 59, 61, 67, 69, 74, 76, 79, 80, 81, 82, 83, 84, 85, 88, 92, 93, 96, 97, 100, 102, 103, 105, 107, 108, 109, 111, 112, 113, 115, 118, 119, 120, 121, 124, 126

Offset

1,1

Comments

Integers n such that the elliptic curve y^2 = x^3 + (n^2+1)*x^2 + n^2*x has a positive rank.

Links

Table of n, a(n) for n=1..64.

Allan MacLeod, Solutions to Leech's problem

Example

a(1)=7 because n=7 is the smallest factor n for which both b^2+h^2 and b^2+(n*h)^2 are squares. The corresponding values of base b and height h are b=A117320(1)=12 and h=A117321(1)=5. 12^2+5^2=169=13^2 and 12^2+(7*5)^2=1369=37^2 are both squares.

Prog

(PARI) { isA117319(n) = ellanalyticrank(ellinit([0, n^2+1, 0, n^2, 0]))[1]; } /* Max Alekseyev, Sep 29 2015 */

Crossrefs

Cf. A117320 (bases), A117321 (heights), A228380.

Sequence in context: A079004 A265311 A192268 * A120645 A168158 A278738

Adjacent sequences: A117316 A117317 A117318 * A117320 A117321 A117322

Keyword

nonn

Author

Hugo Pfoertner, Mar 07 2006

Status

approved