A307894 Hypotenuses of primitive Pythagorean triangles with prime length, having the property that the sum and absolute difference of the shorter legs are both prime numbers.

13, 17, 37, 53, 73, 97, 109, 113, 137, 149, 193, 197, 233, 277, 317, 337, 401, 449, 457, 541, 613, 641, 653, 673, 709, 757, 809, 821, 877, 1009, 1061, 1093, 1117, 1129, 1201, 1289, 1297, 1381, 1481, 1549, 1733, 1873, 1877, 1913, 1933, 1997, 2017, 2053, 2153, 2213, 2221, 2377, 2417, 2437, 2557, 2797

Offset: 1

Author

Torlach Rush, May 03 2019

Keywords

• nonn

Comments

Replacing the shorter legs with the sum and absolute difference of the shorter legs may result in an integer-sided triangle, but this is not always the case. For example, {5,12,13}->{7,13,17} and {7,13,17} are the sides of a triangle. However, {60,91,109}->{31,109,151}, but {31,109,151} are not the sides of a triangle. If the replacement results in such a triangle, then the triangle is a scalene integer triangle (A070112) with sides of prime length, and a(n) is a term of A070081.

Sequence provides x-value of solutions to the equation 2*x^2 = y^2 + z^2, with x, y and z primes. - Lamine Ngom, Apr 30 2022

Examples

			13 is a term because 12 +  5 = 17 and 12 -  5 =  7.
17 is a term because 15 +  8 = 23 and 15 -  8 =  7.
37 is a term because 35 + 12 = 47 and 35 - 12 = 23.
		

Crossrefs

Cf. A070081, A070112.

Subset of A008846.

Subset of A307880.