Primarily square, squarely prime

Algebra Level 3

x 2 − y 2 = 855 419 \large x^2 - y^2 = 855\:419

855 419 855\:419 happens to be a prime number . If x , y > 0 x, y>0 are integers, find the value of x x .


The answer is 427710.

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Arjen Vreugdenhil by Arjen Vreugdenhil , 44, USA

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2 solutions

x 2 − y 2 = ( x + y ) ( x − y ) = 855 419. x^2 - y^2 = (x+y)(x-y) = 855\:419. Since this is a prime number, its only factors are 1 and itself. Therefore x + y = 855 419 , x − y = 1. x + y = 855\:419,\ \ \ \ x-y = 1. We find x = 1 2 ( ( x + y ) + ( x − y ) ) = 1 2 ( 855 419 + 1 ) = 427 710 . x = \tfrac12((x+y) + (x-y)) = \tfrac12(855\:419 + 1) = \boxed{427\:710}.

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Relevant wiki: Applying the Perfect Square Identity

We can use the perfect square identity: x 2 − y 2 = ( x + y ) ( x − y ) = 855419. x^2-y^2 = (x+y)(x-y) = 855419. Since this is a prime number, there are only two factors: 1 and itself. Therefore: x + y = 855419. x + y = 855419. \ \ \ x − y = 1. x - y = 1. \ \ \ Now we add them both: x + y + ( x − y ) = 855420. x + y + (x - y) = 855420. 2 x = 855420. 2x = 855420. x = 427710. x = 427710.

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Moderator note:

Strictly speaking, there are 4 divisors. However, you should explain why we only want the positive divisors.

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