More number theory?

Question

Prove that 10^n + 1 is never a perfect square.

Answer 1

any number mod 9 = 0,1.2.3.4.-1.-2.-3,-4

number ^2 mod 9 = 0,1,4,0, 7

10^n+1 mod 9 = 2 so cannot be a perfect square

Answer 2

If 10^n + 1 = x^2 (say)
then 10^n = x^2 - 1
=> 10^n = (x+1)(x-1).
=> 5^n * 2^n = (x+1)(x-1)
This is the product of two alternate numbers.
At least one of these is a multiple of 5, i.e. should end with 0 or 5. It cannot end with 5, since the other number will be odd too, making the product odd. But the LHS is even (power of 10).
Thus, it has to end with a 0 (i.e. has to be a multiple of 10). COnvince urself henceforth that the other number being an alternate in the sequence of numbers, the product cannot be a power of 10.

Answer 3

you must qualify that n is a positive integer. when n = -1 then of course you get a perfect square.

Answer 4

E=Mc2