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Squares and primes

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, \(4\) is the average of consecutive primes \(3\) and \(5\), and \(9\) is the average of consecutive primes \(7\) and \(11\). \(25\) is not such a perfect square as its "neighboring" primes are \(23\) and \(29\), which average to \(26\).

The list of such perfect squares begins as

\(4, 9, 16, 64, 81, 144, 225, 324, 441, 625, 1089, 1681, 2601, ...\).

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 \(n^{2} \gt 1\) be expressed as the average of two (not necessarily consecutive) primes?

While \(25\) is not the average of two consecutive primes, it is the average of primes \(19\) and \(31\). \(121\) is not the average of consecutive primes, but it is the average of primes \(103\) and \(139\).

(iii) How many perfect squares are the averages of two or more distinct pairs of primes?

\(9\) is the average of both prime pairs \((7,11)\) and \((5,13)\). \(64\) is the average of prime pairs \((61,67), (31,97)\) and \((19,109)\). So if we define \(P(n)\) as the number of distinct prime pairs \((p,q)\) for which \(n^{2} = \dfrac{1}{2}(p + q)\) then, for example, \(P(3) = 2\) and \(P(8) = 3\). With this definition, question (ii) becomes a matter of whether or not \(P(n) \ge 1\) for all \(n \gt 1\), and question (iii) becomes a matter of how many integers \(n \gt 1\) there are such that \(P(n) \ge 2\). Also, is there a maximum value for \(P(n)\) over all integers \(n\)?

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!

Note by Brian Charlesworth
6 months, 1 week ago

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Identifying the largest square seems impossible , but I think we can prove that there are infinite squares ... Chinmay Sangawadekar · 6 months ago

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Very interesting. In the second problem, can we extend the problem to whether every(or an infinite sequence of numbers and if there is not an infinite sequence, then how many numbers can be expressed in that form) number can be expressed as an average of two primes? Svatejas Shivakumar · 6 months ago

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@Svatejas Shivakumar Yes, that extension is in essence the Goldbach conjecture, which is still an open question. As stated in the link, the conjecture is that every even integer \(2n, n \ge 2,\) can be expressed as the sum of two primes \(p,q\), i.e. that \(2n = p + q \Longrightarrow n = \dfrac{p + q}{2}\). This last equation mirrors your extension, i.e., that every integer \(n \ge 2\) can be expressed as the average of two (not necessarily distinct) primes. I believe that we can drop the "not necessarily distinct" qualifier if we restrict ourselves to \(n \ge 4\). My function \(P(n)\) is a measure of the number of Goldbach partitions. (Before I posted this note I had never paid any attention to the Goldbach conjecture, but now that I've made the connection I'm becoming aware of just how much research has been done on this problem and its many variations.)

Goldbach's weak conjecture has a still-to-be-verified proof via Harald Helfgott, (2013). So I guess we could could refer to my questions as a set of "Goldbach's perfect square conjectures". These are weaker than the original conjecture but stronger, (I think?), than the weak conjecture. Brian Charlesworth · 6 months ago

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