Perfect Squares Galore!

Number Theory Level 5

Find the sum of all positive integers m and n such that m , n < 1 0 6 m, n < 10^6 and satisfy the equation:

m 2 + m = 10 ( 8 n 2 + n ) m^2 + m = 10(8n^2 + n)

Note: If you believe that (2,1) is the only solution., then your answer is 2 + 1 = 3.


The answer is 1159930.

John Ashley Capellan by John Ashley Capellan , 23, Philippines

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1 solution

The equation is expressible as the Pell equation: x^2 - 5y^2 = -1 = (4m+2)^2 - (5)(16n+1)^2.

Since, the norm is -1 and the first non-trivial solution is (2, 1), the general solution of the equation is given by: x(n) + y(n) = (2 - sqrt(5))^(2n+1).

By comparison of the solutions for x and y with 4m + 2 and 16n + 1, we can imply that the sum of all solutions (m,n) for m and n is 1159930.

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Can you explain the last line? Specifically, what are the values of m and n?

Calvin Lin Staff - 7 years ago

Sir Calvin, by comparison... all of the solutions of the Pell equation: x^2 - 5y^2 = -1 are of that form x(n) + y(n) = (2 - sqrt(5))^(2n+1). But the requirement is that m and n must be less than 1000000, so the first 6 solutions for the Pell equation are (x,y) : (2,1), (38, 17), (682, 305), (12238, 5473), (219602, 98209), (3940598, 1762289). By comparison for (x, y) = (4m + 2, 16n + 1), (m, n) : (9, 1), (170, 19), (3059, 342), (54900, 6138), (985149, 110143). Adding all m and n satisfying the requirements gives the said answer.

John Ashley Capellan - 7 years ago

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