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## Project Euler 9: Find the only Pythagorean triplet, {a, b, c}, for which a + b + c = 1000

#### Project Euler 9 Problem Description

A Pythagorean triplet is a set of three natural numbers, ab < 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.

#### Analysis

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. These triples are commonly written as (a, b, c), and a typical example is (3, 4, 5); 32 + 42 = 52 or 9 + 16 = 25.
A primitive Pythagorean triple is one in which a, b and c are coprime (gcd(a, b, c) = 1) and for any primitive Pythagorean triple, (ka, kb, kc) for any positive integer k is a non-primitive Pythagorean triple.

Solving this problem for one target sum: a + b + c = 1000 in this case, seemed a bit too trivial but solving for any target sum ≤ 3000 and running thousands of trials in a few milliseconds was a bit more challenging.

This solution uses Dickson’s method for generating Pythagorean triples.
To find integer solutions to a2 + b2 = c2, find positive integers r, s, and t such that $r^2 = 2st$ is a square.
Then: a = r + s, b = r + t, c = r + s + t.
From this we see that r is any even integer and that s and t are factors of $\frac{r^2}{2}$
Example: Choose r = 6. Then $\frac{r^2}{2}=18$.
The three factor-pairs of 18 are: (1, 18), (2, 9), and (3, 6). All three factor pairs will produce triples using the above equations.

s = 1, t = 18 produces the triple [7, 24, 25] because x = 6 + 1 = 7, y = 6 + 18 = 24, z = 6 + 1 + 18 = 25.
s = 2, t = 9 produces the triple [8, 15, 17] because x = 6 + 2 = 8, y = 6 + 9 = 15, z = 6 + 2 + 9 = 17.
s = 3, t = 6 produces the triple [9, 12, 15] because x = 6 + 3 = 9, y = 6 + 6 = 12, z = 6 + 3 + 6 = 15. (Since s and t are not coprime, this triple is not primitive.)

This method finds both primitive (when s and t are coprime) and non-primitive triplets, which are required to solve this problem.

We loop through the even r up to 550 to generate all the triplets necessary to solve different problems. In the case of triples having the same sum this method keeps the maximum product found.

For example, Pythagorean triples (15, 20, 25) and (10, 24, 26) both sum to 60, but their respective products are 7500 and 6240. We keep the maximum, 7500.

This program and method
solves all test cases
on HackerRank

#### Project Euler 9 Solution

Runs < 0.001 seconds in Python 2.7.
Use this link to get the Project Euler 9 Solution Python 2.7 source.

• This code was written to handle thousands of queries on a single run and is why we cache the results of triplet sums from 2 to 3000 in the pn dictionary.