, 2008

Assume that $n^2 - 5^{4m+3} > 0$. If $n^2 - 5^{4m+3} = 1$, then $5^{4m+3} = n^2 - 1 = (n-1)(n+1)$. Since $n$ is obviously even the numbers $n-1$ and $n+1$ must be coprime, which implies $n-1 = 1$ and $n+1 = 5^{4m+3}$. This is not possible. If $n$ is divisible by $5$, then $n^2 - 5^{4m+3} \ge 5$, if $n$ is of the form $n = 5k \pm 1$, then

$n^2 - 5^{4m+3} \equiv 1 \pmod{5}$, and if $n = 5k \pm 2$, then $n^2 - 5^{4m+3} \equiv 4 \pmod{5}$. Since none of the remainders are equal to $2$ or $3$ and $n^2 - 5^{4m+3} > 1$, we conclude that $n^2 - 5^{4m+3} \ge 4$.

Now, assume $5^{4m+3} - n^2 > 0$. This number cannot be equal to either $2$ or $3$. We can see this by considering the remainders modulo $5$, because $5^{4m+3} - n^2$ is either divisible by $5$ (and hence greater than or equal to $5$), or $n$ has the form $n = 5k \pm 1$ or $n = 5k \pm 2$. So, either $5^{4m+3} - n^2 \equiv -1 \equiv 4 \pmod{5}$ or $5^{4m+3} - n^2 \equiv -4 \equiv 1 \pmod{5}$.

Assume there exist $m$ and $n$ such that $5^{4m+3} - n^2 = 1$. We can rewrite the equation as
$$\begin{aligned} n^2 &= 5^{4m+3} - 1 = (5-1)(5^{4m+2} + 5^{4m+1} + \dots + 5 + 1) = \\ &= 4(5^{4m+2} + 5^{4m+1} + \dots + 5 + 1). \end{aligned}$$
This implies that $5^{4m+2} + 5^{4m+1} + \dots + 5 + 1$ is a perfect square. If it is equal to $l^2$, then $(l-1)(l+1) = 5(5^{4m+1} + 5^4 + \dots + 5 + 1)$. Let us consider the expression $5^{4m+1} + 5^4 + \dots + 5 + 1$ modulo $4$. We have
$$5^{4m+1} + 5^4 + \dots + 5 + 1 \equiv 1 + 1 + \dots + 1 + 1 \equiv (4m + 2) \equiv 2 \pmod{4},$$
since there are $4m + 2$ summands in the sum. This sum is therefore divisible by $2$ but not by $4$. So, $(l-1)(l+1)$ is even. On the other hand, $l-1$ and $l+1$ have the same parity, so $(l-1)(l+1)$ is divisible by $4$, a contradiction. The numbers $m$ and $n$ such that $5^{4m+3} - n^2 = 1$ do not exist.

We have shown that $n^2 - 5^{4m+3}$ and $5^{4m+3} - n^2$ cannot be equal to $0$, $1$, $2$ or $3$, so the smallest possible value of $|5^{4m+3} - n^2|$ is $4$ and this value is attained when $n=11$ and $m=0$, for example.