However, today's date is a Pythagorean triple (5/13/12, 5

^{2}+12

^{2}=13

^{2}), so I thought I'd post some math questions regarding Pythagorean triples.

But first, here are some definitions to those who don't know what Pythagorean triples are:

The

**Pythagorean theorem**states that for a

**right triangle**whose sides are of length a, b, and c where c is the longest length (the one opposite the right angle),

**a**.

^{2}+b^{2}=c^{2}If all three of a,b, and c are

**positive integers**, then {a,b,c} is a

**Pythagorean triple**.

Now, in honor of Mother's Day I'll call a Pythagorean triple a

**mother**triple if there is no d greater than 1 such that {a/d, b/d, c/d} is also a Pythagorean triple (equivalently, at least 2 of {a,b,c} are relatively prime). Otherwise, it is a

**child**triple.

Here are the questions:

1. Let the date be of the form xx/yy/zz (ie, two digit year, any can be c (the longest)). What are the next 3 days that are pythagorean triples?

2. How many dates are mother triples given the two digit year? How many total dates are Pythagorean triples?

3. How many total dates are triples given a four digit year?

4. What was the last date that was a pythagorean triple given a four digit year?