Finding Significance in Insignificant Numbers

Sometimes in my examination of the numbers that measure my diurnal age, I come across a number with few entries in the OEIS and all of them inscrutable to my limited intellect. 25197 was a case in point. I wrote about this in an earlier post and what I found in the case of that number was that its square (634888809) contained four consecutive 8's. As such, it belonged to an as yet unidentified sequence of numbers with the common property that their squares contained a sequence of four or more 8's. These numbers are shown below:

This is the sequence that I submitted to the OEIS for approval and I'm still waiting to see it published. This example illustrates that apparently insignificant numbers can contain significance that just needs to be unlocked.

Thus we come to today's number, 25206, that is significant in that it marks a point where there is a balance struck between the number of 4k+1 primes and the number of 4k+3 primes. Prime 617249 marks a point where there are exactly 25206 primes of each sort. While this is clearly significant, the problem is that it's shared with many other nearby numbers as can be seen from this extract from OEIS A092198:

0, 1, 3, 6, 44, 1471, 1472, 1473, 1474, 1475, 1476, 25185, 25187, 25188, 25189, 25190, 25196, 25206, 25211, 25212, 25213, 25214, 25215, 25216, 25217, 25218, 25219, 25222, 25224, 25225, 25251, 25253, 25257, 25258, 25410, 25421, 25426, 25427, ...

Note particularly the gap between 1476 and 25185 (the next member in the sequence). These balance points in the number of 4k+1 and 4k+3 primes clearly exist in clusters. This is the reason that I was looking for something different when examining 25206. It's clear that the number cannot be the sum of two squares because the number factors to 2×3×4201 and there is a 4k+3 prime (3) raised to an odd power (3). However, I wondered if there is an integer solution to the equation \( x^2+2y^2=25206 \) even though there is no integer solution to \( x^2+y^2=25206 \). It turns out there are two, x=158, y=11 and x=38, y=109. This is shown geometrically in the diagram below where the solutions, for positive numbers only, are represented on an ellipse.

The frequency of integer solutions to \( x^2+2y^2=N \) seems about equal to that of  \( x^2+y^2=N \), at least judging by a quick count. For example, over the next twenty integers (25206 to 25216), there are six numbers (25211, 25218, 25219, 25222, 25224, 25225) that can be represented in the form \( x^2+2y^2 \) and six numbers (25209, 25210, 25216, 25220, 25225, 25226) that can be represented in the form  \( x^2+y^2 \). 

From this admittedly quick count, it would seem that about 30% of numbers can be represented as a sum of two squares and another 30% as a sum of a square and twice a square. There is possibly no overlap between the two sets and so it would seem that if a number cannot be represented as a sum of two squares then it may well be possible to represent it as a sum of a square and twice a square. 

This line of enquiry opens up all sorts of intriguing questions such as:

• is it possible to represent a number in both the form \( x^2+y^2 \) and \( x^2+2y^2 \)?
• is it possible to represent all numbers in the form \( ax^2+by^2 \) where \( a \) and \( b \) are positive integers?