Question 1053840
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How many positive integers n make the expression {{{7^n +7^3 + 2 * 7^2}}} a perfect square?
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Calculate {{{7^3 + 2*7^2}}} = 441 and notice that 441 = {{{21^2}}}.

Now your equation is

{{{7^n + 441}}} = {{{m^2}}} for some integer "m",  or

{{{7^n}}} = {{{m^2 - 441}}},  or

{{{7^n}}} = (m+21)*(m-21).

Since "7" is a prime number, both (m+21) and (m-21) must be degrees of 7.

So, ask yourself: what are two degrees of 7 that differs by 42 = (m+21)-(m-21).

The answer is OBVIOUS: these degrees are 7 and {{{7^2}}}, and there are NO others.

So, there is only ONE possible value of "n": n = 3.

The other value n=16 of the other tutor is a MISTAKE.
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