Power and Square

Number Theory Level 3

Find the number of integer solutions of the equation $2^n + n^2 = 260400$ .

The answer is 0.

2 solutions

Rajdeep Brahma
Jun 6, 2018

See n is $congruent$ to 0,1,2,3,4,5,6 mod 7....so $n^2$ is $congruent$ to 0,1,2,4 mod 7. $2^3$ is $congruent$ 1 mod 7...so $2^n$ is $congruent$ t0 1,2,4 mod 7(why?).Now u should be able to solve provided u know 7 divides 260400....:)...hope u liked the problem...

Edwin Gray
Aug 25, 2018

The problem is trivial by letting n = 14 and 15. One is much larger, the other much smaller. Ed Gray

Why pls explain

rajdeep brahma - 2 years, 9 months ago

We can ignore negative $n$ , as these give non-integer results. For non-negative $n$ , the function $2^n+n^2$ is clearly strictly increasing (both terms get larger as $n$ increases) and continuous. So there can be at most one solution; @Edwin Gray (and the intermediate value theorem) showed this solution must be between $14$ and $15$ - but there are no such integers.

Chris Lewis - 1 year, 8 months ago

Power and Square

The answer is 0.

2 solutions

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