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


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4 replies to this topic

#1 EventHorizon

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Posted 14 May 2012 - 01:08 AM

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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#2 EventHorizon

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Posted 14 May 2012 - 04:21 AM

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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#3 bonanova

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Posted 14 May 2012 - 04:23 AM

Start the ball rolling.

Spoiler for next three

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#4 bonanova

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Posted 14 May 2012 - 04:41 AM

Total PT dates with 2-digit years.

Spoiler for looks like

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#5 lionsbru

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Posted 08 December 2013 - 10:49 AM

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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