G. H. Hardy tells the following story about visiting his colleague Ramanujan.
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
This story has become famous, but the rest of the conversation isn’t as well known. Hardy followed up by asking Ramanujan what the corresponding number would be for 4th powers. Ramanujan replied that he did not know, but that such a number must be very large.
Hardy tells this story in his 1937 paper “The Indian Mathematician Ramanujan.” He gives a footnote saying that Euler discovered 635318657 = 158^4 + 59^4 = 134^4 + 133^4 and that this was the smallest number known to be the sum of two fourth powers in two ways. It seems odd now to think of such questions being unresolved. Today we’d ask Hardy “What do you mean 635318657 is the smallest known example? Why didn’t you write a little program to find out whether it really is the smallest?”
Surely someone has since written such a program and settled the question. But as an exercise, imagine the question is still open. Write a program to either come up with a smaller number that answer’s Hardy’s question, or prove that Euler’s number is the smallest one. To make the task more interesting, you might see whether you could do a little pencil-and-paper math up front to reduce the amount searching needed. Also, you might try writing the program in different styles: imperative, functional, etc.