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Summary: Math 250: Pythagorean Triples
We all are familiar with the Pythagorean Theorem, which states that the lengths a and b of the legs
of a right triangle, together with the length c of the hypotenuse, satisfy the relation a2
+ b2
= c2
.
The triple (a, b, c) is a Pythagorean Triple iff a, b, and c satisfy the Pythagorean Theorem. A
Pythagorean triple is primitive iff a, b, and c have no common integer divisor except 1. A pair of
numbers p and q is relatively prime iff they have no common integer divisor except 1. Note that
the converse of the Pythagorean Theorem is also true.
Throughout the discussion below, we need to specify the parity of an integer n, namely whether
it is even (n = 2j for some integer j) or odd (n = 2k + 1 for some integer k). We will let N be the
set of natural numbers {1, 2, 3, · · · }, and Z the set of integers {0, ±1, ±2, · · · }.
1. Prove that p is odd iff p2
is odd, and q2
is even iff q is even. Hint: To show p2
odd = p odd,
consider p2
- 1.
2. If n Z, then n2
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