Largest N

Number Theory Level 4

What is the largest integer n n for which n 2 + 66 n − 959 n^2 + 66n-959 is a perfect square ?


The answer is 480.

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Aira Thalca by Aira Thalca , 24, Indonesia

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1 solution

Kushal Bose
Dec 27, 2016

n 2 + 66 n + 959 = m 2 = > n 2 + 2.33 n + 3 3 3 − 3 3 2 − 959 = m 2 = > ( n + 33 ) 2 − 2048 = m 2 = > ( n + 33 ) 2 − m 2 = 2048 = > ( n + m − 33 ) ( n − m + 33 ) = 2048 = 2 11 n^2+66n+959=m^2 \\ =>n^2 +2.33n+33^3-33^2-959=m^2 \\ =>(n+33)^2-2048=m^2 \\ =>(n+33)^2-m^2=2048 \\ =>(n+m-33)(n-m+33)=2048=2^{11}

For maximum value of n n first should be maximum and second part should be minimum Both of these values should be even As m + n + 33 > m − n + 33 m+n+33 >m-n+33 So, for the largest value of n n we have to assign values

n + m + 33 = 1024 n+m+33=1024 and n − m + 33 = 2 n-m+33=2 .Adding these two we get n = 480 n=480

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Good solution.

For completeness, you should explain why
1. "for maximum value of n, first should be maximum ..."
2. We didn't use n + m + 33 = 2048 , n − m + 33 = 1 n + m + 33 = 2048, n - m + 33 = 1 which yields a larger first part.

Calvin Lin Staff - 4 years, 5 months ago

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Two values should be even that's why n − m + 33 = 2 n-m+33=2

Kushal Bose - 4 years, 5 months ago

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For completeness, statements/observations like this should be stated in the solution directly, and not left as a test of mindreading abilities.

Calvin Lin Staff - 4 years, 5 months ago

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