A268603 Denominator of the side lengths (legs in ascending order) of the easiest Pythagorean Triangle (with smallest hypotenuse) according to the congruent numbers A003273.

2, 3, 6, 1, 1, 1, 12, 5, 60, 323, 30, 9690, 3, 2, 6, 1, 2, 2, 1, 3, 3, 2, 1, 2, 35, 3, 105, 20748, 3485, 72306780

Offset: 1

Author

Martin Renner, Feb 08 2016

Keywords

• nonn
• frac
• tabf
• more

Comments

Every three fractions x < y < z satisfy the Pythagorean equation x^2 + y^2 = z^2: (A268602(3*n-2)/a(3*n-2))^2 + (A268602(3*n-1)/a(3*n-1))^2 = (A268602(3*n)/a(3*n))^2.

The area A = x*y/2 of these Pythagorean triangles is a congruent number: A003273(n) = (1/2) * (A268602(3*n-2)/a(3*n-2)) * (A268602(3*n-1)/a(3*n-1)).

Examples

			The first congruent number is 5 and the associated right triangle with the side lengths x = 3/2, y = 20/3, z = 41/6 satisfies the Pythagorean equation (3/2)^2 + (20/3)^2 = (41/6)^2 and the area of this triangle equals 1/2*3/2*20/3 = 5.
		

Links

• Eric Weisstein's World of Mathematics, Congruent Number.

Crossrefs

Cf. A003273, A268602.

Extensions

a(14) corrected on Mar 14 2020