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Happy Pythagorean Triple Day!! Also, Mother's Day.



Yes, I know Mother's Day is the much bigger deal (Happy Mother's Day to all mothers out there! I hope you feel loved and appreciated every day of the year.).

However, today's date is a Pythagorean triple (5/13/12, 52+122=132), 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), a2+b2=c2.

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?

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4 answers to this question

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I was rushing out the door when I posted this thread, so here's a quick clarification.

When I say total dates, I meant both mother and child triples included. 'total' should have been replaced with the words 'either mother or child' or 'any type of' in front of 'Pythagorean triple' in both question 2 and 3.

For question 2, I mean per century (ie, ignore wrap around, which would make the answer infinite).

For question 3 and 4, I meant any number of digits for the year, so it never wraps around to 0. For question 4 it makes the answer the last one that will ever happen.

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Total PT dates with 2-digit years.

Constraints would be one number < 13, one < 32 (maybe smaller) and one < 100.

3/4/5 (6 permutations)

5/12/13 (4 permutations, 13 can't be the month)

7/24/25 (2 permutations, only 7 can be the month)

8/15/17 (2 permutations, only 8 can be the month)

12/35/37 comes close, but no cigar.

Total of 14.

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The previous answer lists only PRIMITIVE Triples. Unless I've misread the rules, I see no cause to omit non-primitive triples, in which case we may add:
6/8/10 (6 permutations)
9/12/15 (4 permutations, 15 can't be the month)
12/16/20 (2 permutations, only 12 can be the month)
This yields an additional 12 for a total of 26.
For what it's worth, if we follow those who limit Pythagorean Triple DATES to those in which only "C" can be the year, the number is reduced to the following 11:
3/4/5, 4/3/5
6/8/10, 8/6/10,
5/12/13, 12/5/13
9/12/15, 12/9/15
8/15/17, 12/16/20, 7/24/25

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