Proof of Theorem and Solutions to Equation (3.2)

If any triple of integers (x, y, z) satisfy (3.2), then so do any of (+x, ±y, +z); thus, to show the theorem, it suffices to show that there are 110 positive integer solutions to (3.2). The proof is accomplished by showing that if some solution (r, y, z) to (3.2) exists, then from that solution one can create another "smaller" solution (a', y', 2), where "smaller" means that z' < z. Since the positive integers are well-ordered, this process cannot continue forever, and so one must abandon the original assumption that some solution exists.

Modulo 4, x and y cannot both be odd.

Let x and y be even, say x = 2k, y = 2E; then 16 divides z², and so 4 divides z, say z = 4m.

Then (2k)4 + (tn)2, and division by 16 yields k4 + = in2, a smaller solution.

Similarly, if any prime p divides both x and y, write x = pk, y = p€, and z = p2m. Division by p4 shows that a? = k, = €, and a' = in is another smaller solution.

Hence, it suffices to assume that x and y are relatively prime.

Suppose that x is even and y is odd.