Square Expression!

Number Theory Level 4

Find the sum of all integers $n$ for which $n^2+n+41$ is a perfect square.

The answer is -1.

1 solution

Ratul Pan
Sep 18, 2015

Let $n^2$ +n+41= $m^2$
or, 4 $m^2$ =4 $n^2$ + 4n+ 164 (multiplying by 4)
or, 4 $m^2$ =( $2n+1)^2$ +163
or, (2m+2n+1)(2m-2n-1)=163
since 163 is prime, therefore we must have
2m+2n+1=(+-)1, (+-)163.....................i
2m-2n-1=(+-)163, (+-)1......................ii
subtracting (i-ii)
4n+2=(-+)162
hence n=-41, n=40
therefore sum=-41+40=-1

Ratul bhai ratul,topper solution by topper guy

Subhra Patra - 3 years ago

Square Expression!

The answer is -1.

1 solution

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