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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.

Have $a + b + c = 1000$ and $a^2 + b^2 = c^2$

We can reduce this with a little algebra:

$$a^2 = c^2 - b^2 = (c + b)(c - b)$$

So $$a^2 = (1000 - a)(1000 - a - 2b)$$

Simplifying: $$ a + b - \frac{ab}{1000} = 500 \\ b = \frac{1000 (500 - a)}{1000 - a} $$



In [7]:

    
for i in range(2, 500):
    if 1000*(500 - i) % (1000 - i) == 0:
        a = i
        b = (1000*(500 - i))/(1000 - i)
        c = 1000 - a - b
        break
print "a:",a, "b:",b, "c:",c
print a**2 + b**2, c**2
print a*b*c









    



a: 200 b: 375 c: 425
180625 180625
31875000



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