23. The End of the World. On April 1, 1946, the newspaper "Jericho Daily Lapse" reported: "A famous astrologer and numerologist from Guyasuela, Professor Euclid Paracelsus Bombast Umbugio, predicted that the end of the world will occur in 2141. His prediction is based on deep mathematical and historical research. Professor Umbugio calculated the expression

$$
1492^{n}-1770^{n}-1863^{n}+2141^{n}
$$

for \( n=0,1,2,3 \) and so on up to 1945, and found that all the numbers he obtained after months of painstaking work are divisible by 1946. The numbers 1492, 1770, and 1863 represent historical dates: the discovery of the New World, the Boston Massacre, and the Gettysburg Address, respectively. What could happen in 2141? Obviously, the end of the world."

We should strip such a professor of his title! Obtain his result with a simple calculation.
23. All that is needed to solve the problem is that $x^{n}-y^{n}$ for $n=0,1,2, \ldots$ is divisible by $x-y$. Let the quantity the professor was calculating be $F(n)$. Then, since $2141-1863=$ $1770-1492=278, F(n)$ is divisible by 278 for any $n$. Similarly, $2141-1770=1863-1492=371$, a number coprime with 278. Thus, $F(n)$ is always divisible by $278 \cdot 371=53 \cdot 1946$ and, of course, by the number 1946 itself.

[E. P. Stark, A. M. M., 54, 43 (January 1947).]