#WeHateMath Project Euler 9 – Pythagorean Triplets

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

For example, 32 + 42 = 9 + 16 = 25 = 52.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.

Find the product abc.

Let’s get the problem part out of the way first. It turns out to be pretty easy to generate Pythagorean Triplets (sometimes called Pythagorean triples):

Pick two positive integers, m and n, ,where m < n. Generate your three values like so:

Of course, doing this by hand is annoying and this is just the sort of thing for which spreadsheets were made. I set up the formulas and then starting fiddling with values of m and n and it just took a few tries to range in on the answer:

m

5

n

20

a

375

b

200

c

425

Sum(a,b,c)

1,000

Product(a,b,c)

31,875,000

a^2

140,625

b^2

40,000

a^2 plus b^2

180,625

c^2

180,625

Readers know by now that the point of the Project Euler posts is not to simply get the answer to the problem but to figure out the point of the problem. In this case, that’s not obvious. Though Pythagorean Triples are a surprisingly hot topic of discussion among mathematicians if my casual research is anything to go by. I even found a paper documenting a new way to generate triples that suggests an interesting application of the math to cryptology. It’s especially surprising when you consider that these triples have been around for over 2000 years.

The classic Pythagorean triples are (3, 4, 5) and (5, 12, 13). You can generate new triples by using multiples of each element. In our answer above, (375, 200, 425) is a multiple of (3, 4, 5). There are others, however. For example (4, 3, 5) is also a valid triplet but can’t be generated by the traditional methods.

Pythagorean Triples are connected to both Fibonacci numbers and primes, which we’ve discussed previously.. The two classic triples each have Fibonacci values as two of the three values. In addition, up to two of the legs can be prime numbers but because at least one of the legs must be even, there is no triple where all three elements are primes.

This is a great example of pure mathematics that may (or may not) someday find some practical application. I view mathematics as a way to perform experiments on the universe, but unlike traditional laboratory science, my lab bench is conveniently located between my ears. So the next time someone tells you that studying math is impractical, remind them:


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