Find the smallest positive integer $k$ which is representable in the form $k=19^{n}-5^{m}$ for some positive integers $m$ and $n$.
Answer: 14.

Assume that there are integers $n, m$ such that $k=19^{n}-5^{m}$ is a positive integer smaller than $19^{1}-5^{1}=14$. For obvious reasons, $n$ and $m$ must be positive.

Case 1: Assume that $n$ is even. Then the last digit of $k$ is 6 . Consequently, we have $19^{n}-5^{m}=6$. Considering this equation modulo 3 implies that $m$
must be even as well. With $n=2 n^{\prime}$ and $m=2 m^{\prime}$ the above equation can be restated as $\left(19^{n^{\prime}}+5^{m^{\prime}}\right)\left(19^{n^{\prime}}-5^{m^{\prime}}\right)=6$ which evidently has no solution in positive integers.

Case 2: Assume that $n$ is odd. Then the last digit of $k$ is 4 . Consequently, we have $19^{n}-5^{m}=4$. On the other hand, the remainder of $19^{n}-5^{m}$ modulo 3 is never 1 , a contradiction.