Fermat vs. Pythagoras

Computer generated and assisted proofs and verification occupy a small niche in the realm of Computer Science. The first proof of the four-color problem was completed with the assistance of a computer program and current efforts in verification have succeeded in verifying the translation of high-level code down to the chip level.

This problem deals with computing quantities relating to part of Fermat's Last Theorem: that there are no integer solutions of for n > 2.

Given a positive integer N, you are to write a program that computes two quantities regarding the solution of


where x, y, and z are constrained to be positive integers less than or equal to N. You are to compute the number of triples (x,y,z) such that x<y< z, and they are relatively prime, i.e., have no common divisor larger than 1. You are also to compute the number of values 0<p≤n such that p is not part of any triple (not just relatively prime triples).

The input consists of a sequence of positive integers, one per line. Each integer in the input file will be less than or equal to 1,000,000. Input is terminated by end-of-file.

For each integer N in the input file print two integers separated by a space. The first integer is the number of relatively prime triples (such that each component of the triple is <=N ). The second number is the number of positive integers <=N that are not part of any triple whose components are all <=N . There should be one output line for each input line.

10
25
100

1 4
4 9
16 27

UVA