# Project Euler Problem 9 in F#

Project Euler Problem 9 introduces the interesting mathematical concept of Pythagorean Triple. Let's have a look at the question:

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
a^2 + b^2 = c^2

For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

As usual, I first show my solution and then comment on the approach:

Let's go through the solution step by step:

• The first function, pythagorean_triplets uses Euclid's Formula to enumerate all possible Pytagorean Triplets up to a given threshold. The formula can be summarized like this:
$$a = m^2 - n^2, b = 2mn, c = m^2 + n^2$$, where $$m$$ and $$n$$ are positive integers with $$m > n$$.

It's interesting to note that each item of the list is itself a list of three numbers. I could have generated a tuple, but since in the end I need to multiply all the 3 values together, the list was more straightfoward to use.

• multiply_list is just a convenience function to multiply all the elements of a list together
• find_triplet_with_sum does the heavy work of generating the actual triplets picking the first one where the sum is equal to a given value.
• At the end I just put the pieces together and solve the problem.

One caveat: Euclid's Formula does not generate all Pytagorean Triplets(there are other formulas that do). I can say that I have been lucky that it generated the one requested by the Project's Euler problem. The code above might not work if the goal is changed to find a triplet with a different product.

Updated: