7^n+24^n is perfect square

[kasd]

Find all n-s where n belongs to N and n>1 and 7^n+24^n is perfect square. I have no idea where to start.
I'm thinking of looking at two cases:
1. n=2*k
2. n=2*k+1
but i have no idea what to do
the answers are 2 and 25

 

[Deleted member 4993]

Find all n-s where n belongs to N and n>1 and 7^n+24^n is perfect square. I have no idea where to start.
I'm thinking of looking at two cases:
1. n=2*k
2. n=2*k+1
but i have no idea what to do
the answers are 2 and 25

Your problem states:

"Find all n-s where n belongs to N and n>1..."

What is "s"?​

​

Where is "s"?​

 

[Harry_the_cat]

I think it is plural of n.

 

[Cubist]

Find all n-s where n belongs to N and n>1 and 7^n+24^n is perfect square.
...
the answers are 2 and 25

2 works, but 25 doesn't...

          7^25+24^25 = 32009658644408160055397619313151431  this close to being a square, but it isn't a perfect square...
178912432895000788^2 = 32009658644408160047019242520620944

 

[Cubist]

this close to being a square, but it isn't a perfect square...

Correction:- I should have said "this is close to being a perfect square, but no cigar!" That's because the number is a square of some real amount, say "x", but x won't be an integer

 

[Steven G]

I would try to use mod7 and mod 24.

 

[Steven G]

Your problem states:

"Find all n-s where n belongs to N and n>1..."

What is "s"?​

​

Where is "s"?​

n-s means numbers.

 

[Deleted member 4993]

n-s means numbers.

welcome back Steve....

 

[Steven G]

Thank you!

 

[AmandasMathHelp]

Getting back to the problem, I have found something interesting. 7^n can only end in the numbers 9,3,1,7 and 24^n can only end in numbers 4,6. And this happens in predictable ways ! 24^n ends in 4 when n is odd and 6 when n is even. Also for 7^2 you can figure out the ending number based on n mod 4. Also any number squared can only end in 0,1,4,5,6,9. So if you look at different possibilites for what the ending number of 7^n+24^n is under different n mod 4s, you can see if that number is even possible to be a perfect square based on its last digit. It eliminates a lot of possibilities, but I have't found a way to find real solutions.

Also btw to the other commentors, I think the 25 is the number that is squared (and thats why it is included in the answer). [MATH] 7^2+24^2=25^2[/MATH]