RMO Problem

Number Theory Level 4

y 2 = x 5 − 1 x − 1 , x ≠ 1 \large y^2 = \dfrac{x^5 -1}{x-1}, \quad x \ne 1

Let the positive integer solution of the equation above be ( m , n ) (m, n) . Find m + n m+n .


The answer is 14.

Abdullah Ahmed by Abdullah Ahmed , 22, Bangladesh

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1 solution

Chaebum Sheen
Dec 25, 2016

Note that 4 y 2 = 4 x 4 + 4 x 3 + 4 x 2 + 4 x + 4 4y^2= 4x^4+4x^3+4x^2+4x+4

Then, we can use that if x > 3 x > 3 and k = 2 y k=2y

4 x 4 + 4 x 3 + x 2 = ( 2 x 2 + x ) 2 < k 2 < 4 x 4 + 4 x 3 + 5 x 2 + 2 x + 1 = ( 2 x 2 + x + 1 ) 2 4x^4+4x^3+ x^2= (2x^2+x)^2 < k^2 < 4x^4+4x^3+5x^2+2x+1 = (2x^2+x+1)^2

So we have that x x is either 3 3 or 2 2 . Trial and error shows that ( x , y ) = ( 3 , 11 ) (x,y)=(3,11)

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