SOLUTION: How many positive integers n make the expression {{{7^n +7^3 + 2 * 7^2}}} a perfect square?

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Question 1053840: How many positive integers n make the expression a perfect square?
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
How many positive integers n make the expression a perfect square?
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IDK
3 and 16 work. That's 2.
Might be others.
Answer by ikleyn(52879)   (Show Source): You can put this solution on YOUR website!
.
How many positive integers n make the expression a perfect square?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Calculate  = 441 and notice that 441 = .

Now your equation is

 =  for some integer "m",  or

 = ,  or

 = (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 , 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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