SumOf2Squares and Random Primes


This post/article is a followup to a previous post about the Sum Of 2 Squares and how none of the terms of the formula can share prime factors. It’s required reading (but you’ll like it).


When last we left off, we had been discussing that:

1/2 of all primes are of the “SumOf2Squares” type (aka “SOTS” type”)


This ratio of 1/2 is seemingly extremely consistent

across all ranges of any significant size!


Anyway, so why are 1/2 of the primes seemingly of the “SumOf2Squares” type (aka “SOTS” type”)? Here is my conjecture (“explanation”). Follow me on this…


First of all, we know that a “SOTS prime” P is a prime that can be expressed as the sum of 2 squares; like so…

P = X2 + Y2And we also know that for this to be possible then (P – 1) must be divisible by 4. For example:

13 – 1 = 12 and 12 is divisible by 4 so 13 is a “SOTS prime” which means we can express 13 (P) as the sum of 2 squares; like so…

13 = 32 + 22

With that in mind, we might also observe that starting with 8, every other EVEN integer is divisible by 4. For example:

8, 12, 16, 20, 24, … are all divisible by 4.


From this we realize that every other ODD integer O will satisfy the first SOTS requirement of (O – 1) is divisible by 4

For example 9, 13, 17, 21, … 9-1 is divisible by 4, 13-1 is divisible by 4, and so on.


However, actually being a prime is the other/second requirement of being a “SOTS prime” and we can see that not every other ODD integer is a prime! For example, neither 9 nor 21 from the above list is a prime. But, what we’re after here is “explaining” why 1/2 of the primes are SOTS primes.


The Effect Of Random Distribution of Primes

Now, IF the primes are randomly distributed among the odd integers then we could expect as many primes P would randomly “land” on an ODD number that fullfilled the first requirement as would not. That is, it would be as likely as not that a prime P would “land” on an ODD number that met the requirement of (O – 1) being divisible by 4.   This would explain why, seemingly, and as conjectured in the prior post/article, that 1/2 of the primes are seemingly of the “Sum Of 2 Squares type.” That is, why 1/2 of the primes can be expressed as the Sum Of 2 Squares.

When I say “seemingly” it’s because it’s based on experimental evidence using the PrimeTest.exe program (see the prior post). Whew!.

It is generally thought that primes are “sort of” randomly distributed along the number line but within the “fact” that they “thin out” according to the Prime Number Theorem (PNT). Or put another way, within any significant/sizable range they are “kinda sort of pretty much” distributed randomly. We don’t know whether, or under what conditions, this “sort of nearly” random distribution falls apart. For example, does “nearly random” distribution of primes fall apart completely after 101234567890123456789? Or, conversely, could all primes greater than 101234567890123456789 be SOTS primes? Who knows. That said, at the bottom of this post are links to related articles that you may want to read.


So here is my first conjecture:

Due to the “pretty much” random distribution of primes then “pretty much” 1/2 of all primes will be SOTS primes..

The above first conjecture is pretty strong especially when using the term “pretty much.” It would probably (“pretty much”) not garner much support at the next AMS conference so let’s try this conjecture instead:


Due to the “random enough” distribution of primes even as we go to infinity, then we can say there are an infinite number of SOTS primes (an infinite number of primes that can be expressed as the Sum Of 2 Squares)..

I’m going to go with this last (second) conjecture and I’m sticking to it… but my gut tells me both are true.


The End…Except for the interesting links below.



There are many more articles to be found by searching on “random distribution of primes” or just “distribution of primes.”


New Pattern Found In Prime Numbers


Peculiar Pattern Found in “Random” Prime Numbers – Last digits of nearby primes have “anti-sameness” bias


Prime number theorem

Structure and randomness in the prime numbers


The End