There are many mathematical puzzles, some with answers, some without, and some that raise interesting mathematical or philosophical questions, while others are trivial or even trollish. I will let the reader decide where the following puzzle, and my answer to it lies as far as truth and beauty go.
What is the average distance between consecutive prime numbers? According to one of the oldest and most elegant proofs in mathematics, there are an infinite number of prime numbers. This means that the gap between prime numbers grows indefinitely large, but it can be infinite. The same fact of infinite primes means that the mean distance between primes can't be infinite, and yet it has to be indefinitely large. It almost would lead to the creation of a new mathematical concept: a subinfinity, a number that is indefinitely large, but still by definition smaller than infinity. The mind boggles at the concept. Take a minute to savor the antinomy, before, in the next paragraph, I reveal the real answer.
The real answer is two. 2. One plus one. How can this be?
The original question was phrased to ask the average. But average can mean many things. It usually means the mean. But there is another definition of average: the mode. The answer that occurs most frequently. It is currently not known whether there is an infinite amount of twin primes, but even if this is never formally proved, there are a great number of twin primes, and a gap of two is the most common gap between consecutive prime numbers. Therefore, the easiest answer to this vexing question of indefinite versus infinite is merely to choose a different definition of average, and then the answer is "two".