What was Euler's solution to the prime number problem, and why is it considered so elegent and beautiful?

Answer 1

The following three links are the best I found for your question, which isn't exactly on target to begin with. Euler did not "solve the prime number problem". There is no one particular "prime number problem", but there is certainly the "prime number theorem", which is famously tied in with the Riemann Hypothesis, one of the very most important in mathematics. The links explain the connection and Euler's role. The PDF article (1st one) is the best, but you need an Adobe Acrobat Reader.