A few questions I've been wondering about regarding primes and squares that I thought the Brilliant community might have some insight on ...
(i) Are there an infinite number of perfect squares that are the averages of consecutive primes?
As examples, is the average of consecutive primes and , and is the average of consecutive primes and . is not such a perfect square as its "neighboring" primes are and , which average to .
The list of such perfect squares begins as
It becomes less and less likely as the squares get larger that they will be the average of successive primes, but the notion that there is a largest such square seems unlikely. So the problem is to either prove that there is no such largest square, or prove that there must be one and in fact identify it.
(ii) Can every perfect square be expressed as the average of two (not necessarily consecutive) primes?
While is not the average of two consecutive primes, it is the average of primes and . is not the average of consecutive primes, but it is the average of primes and .
(iii) How many perfect squares are the averages of two or more distinct pairs of primes?
is the average of both prime pairs and . is the average of prime pairs and . So if we define as the number of distinct prime pairs for which then, for example, and . With this definition, question (ii) becomes a matter of whether or not for all , and question (iii) becomes a matter of how many integers there are such that . Also, is there a maximum value for over all integers ?
I'm not sure if these are open questions or have in fact been solved centuries ago, so I thought I might learn a few things by sharing them with the community. Enjoy!