A206334 Numbers n such that there is a triangle with area n, side n, and the other two sides rational.

3, 5, 7, 10, 12, 15, 16, 18, 19, 23, 25, 26, 27, 28, 29, 30, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 55, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 95, 96, 97, 103, 104, 105, 106, 107, 109, 115, 119, 122, 123, 124, 125, 126

Offset: 1

Author

James R. Buddenhagen, Feb 06 2012

Keywords

• nonn

Comments

n>3 is in the sequence just in case the elliptic curve y^2 = 4*x^4 + (n^2+8)*x^2 + 4 has positive rank. Note that (0,2) is on that curve.

n is in the sequence just in case there are positive rational numbers x,y such that x*y>1 and x - 1/x + y - 1/y = n.

The triangle whose sides are [(4*k^6+8*k^5+8*k^4+4*k^3+2*k^2+2*k+1)/((k+1)*k*(2*k^2+2*k+1)), (4*k^6+16*k^5+28*k^4+28*k^3+18*k^2+6*k+1)/((k+1)*k*(2*k^2+2*k+1)), 4*k^2+4*k+4] has area equal to its third side. Hence, starting with the second term, A112087 is a subsequence of the present sequence.

The triangle whose sides are [(k^6+2*k^4+k^2+1)/(k*(k^2+1)), (k^4+3*k^2+1)/(k*(k^2+1)), (k^2+2)*k] has area equal to its third side. Hence, starting with the first positive term, A054602 is a subsequence of the present sequence. [This subsequence found by Dragan K, see second link, below.]

The triangle whose sides are [(k^8+6*k^6+13*k^4+13*k^2+4)/(k*(k^2+2)*(k^2+1)), (k^6+3*k^4+5*k^2+4)/(k*(k^2+2)*(k^2+1)), k*(k^2+4)] has area equal to its third side. Hence A155965 is a subsequence of the present sequence.

Examples

			5 is in the sequence because the triangle with sides (37/6, 13/6, 5) has area 5, one side 5, and the other two sides rational.
		

Links

• James R. Buddenhagen, Table of triangles up to n = 145
• Dragan K and Rita the dog, Question and answer [broken link]
• Ian Connell, APECS elliptic curve software (which runs under old versions of Maple).
• Eric Weisstein's World of Mathematics, Heronian Triangle.

Crossrefs

Cf. A112087, A054602, A155965, and A206351 (subsequences, see comments).