# Understand the problem

##### Source of the problem

##### Topic

##### Difficulty Level

##### Suggested Book

# Start with hints

Let’s get an idea of m and n by using the modulo technique.

To get an idea of n, we must remove or eliminate m, to do that we take modulo 10.

Observe that the equation demands to be taken modulo 10, given the numbers and it turns out that \( 19^n = (-1)^n = 1 mod 10 \). It implies that n must be even.

Try to get an idea of m now.

Also, (0,0) is a solution. So, we take both m and n as non-zero.

To remove n, using the information that n is even, we can remove the variable n, taking modulo 4.

So, \( 2m^2 + 1 = 1 mod 4 \) implies m must be even. Let m = 2p.

Observe that RHS is a square and LHS \( < 20^{2p} \).

Thus, \( LHS \leq (20^{p} – 1)^2 \).

Hence ,

which simplifies to . (*)

This gives the only solution (2,2) as (m,n).

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