Prime Numbers & Sum of Squares

I am a hobbyist mathematician with no formal coursework in Number Theory. And I am not associated with a university, so I don’t have free access to the publications on number theory.

But I do have a degree in Physics, and one of the few things on Earth more arrogant than a mathematician is a physicist. So, I fearlessly play with ideas in number theory, with the understanding that 99.9% of the time I am discovering things that are already known.

This thread is mostly concerned with X = a² + b²; when both a,b are positive integers, and, for certain forms of X, are prime with surprisingly high-frequency. I’ll be documenting the work I’ve done and welcome on-topic comments and on-topic discoveries of your own.

A particular ‘Divine’ number

Let’s recall that the list of noncomposite numbers that are <50 is:


This list of sixteen numbers is all of the fifteen prime numbers <50 and 1. Of course, 1 is a nonprime, noncomposite number.

And now let me introduce the most amazing number you have never heard of. If you enjoy the subject of prime numbers then this particular number will leave you slack-jawed in disbelief, because this number gives rise to a large fraction of the numbers on the above list. I will define the term “divine number” later in this thread. But I introduce you to a particular divine number here, namely:



Of course, 13³ + 13 = 2210. And, at first glance, 2210 seems like a very pedestrian integer, only remarkable because of its ordinariness and lack of any interesting properties. But now let’s blow your mind.



Okay, stare at this list below, particularly at the unbolded numbers; what association could they possibly have with 2210?



The big reveal:



Also, this:



Let’s have a drum roll, please. For your entertainment:



And finally,

Now, I ask you, have you ever before looked at the list of primes and realized that that so many of the low primes were related in this way?

My definition of a so-called ‘divine” number, X, is a composite number that has two or more solutions to X = a² + b², in which a,b are positive integers, (a<b), and for which all of the values of a and b are prime numbers. Given this definition, the number 2210 is “divine.”

Weird sidenote
This post is unrelated to the main topic of this thread but does provide one additional remarkable quality of the number 2210.

Note that we can define 2210 as the following product:


Now let’s take each of the first four factors in the above equation and put them into the general expression (n³ + n.)









The fact that each of the four first factors, when defined as n in the expression (n³ + n), produces a sub-product of 2210 (or the number 2210 itself) seems remarkable to me. I wonder whether any other number has a property even remotely similar to this.

Alas, this sequence does not continue because:

.

I have played around with but don’t see any clear connection between this property and the ability of 2210 to be associated with so many noncomposite numbers in the range 1 to 47. So, this weird sidenote seems to belong in a box labeled “interesting but not relevant.”

Of course, one may take any two prime numbers (or 1 and a prime number) and take the sum of their squares to produce a composite number; thus there are an infinite number of composite integers, X, such that


A topic I have been investigating is composite numbers which have two or more pairs of numbers that are a solution to

and for which all of the pairs that are solutions are comprised of distinct integers that are either 1 or prime numbers. I have been defining such numbers as divine composite numbers

Examples
The lowest divine composite number is 170 which is the sum of the squares of two pairs of numbers: (1,13) and (7,11), the elements of which are all 1 or prime.

The number 50 is the sum of the squares of two pairs of integers: (1,7) and (5,5) but does not qualify as a divine number because the (5,5) pair is not compromised of two distinct integers.

We earlier characterized 2210 as being equal to the sums of squares of 4 pairs of numbers: (1,47); (19,43); (23,41) and (29,37). All four of those pairs are compromised of distinct prime numbers (or in the case of (1,47) compromised of 1 and a prime.) Thus 2210 is the smallest number to be 4-fold divine.

Here are some more divine number with four sets of prime/noncomposite pairs:

5330=2*5*13*41
= 1² + 73²
= 17² + 71²
= 29² + 67²
= 43² + 59²

11570=2*5*13*89
= 11² + 107²
= 31² + 103²
= 37² + 101²
= 73² + 79²

13130=2*5*13*101
= 19² + 113²
= 41² + 107²
= 61² + 97²
= 79² + 83²

17810=2*5*13*137
= 11² + 133²
= 41² + 127²
= 71² + 113²
= 77² + 109²

19370=2*5*13*149
= 7² + 139²
= 41² + 133²
= 47² + 131²
= 89² + 107²

124610=2*5*17*733
= 1² + 353²
= 53² + 349²
= 167² + 311²
= 211² + 283²

237170=2*5*37*641
= 1² + 487²
= 151² + 463²
= 157² + 461²
= 293² + 389²

246410=2*5*41*601
= 73² + 491²
= 179² + 463²
= 263² + 421²
= 349² + 353²

223130 = 2*5*53*421
= 71² + 467²
= 103² + 461²
= 307² + 359²
= 331² + 337²

574010 = 2*5*61*941
= 31² + 757²
= 167² + 739²
= 479² + 587²
= 491² + 577²

39338 = 2*13*17*89
= 23² + 197²
= 97² + 173²
= 107² + 167²
= 113² + 163²

39962 = 2*13*29*53
= 19² + 199²
= 59² + 191²
= 89² + 179²
= 131² + 151²

103298 = 2*13*29*137
= 53² + 317²
= 73² + 313²
= 163² + 277²
= 193² + 257²

I have found more 4-fold divine numbers and can post them if there is interest.

As may be evident from the above examples, in-order for a number to be 4-fold divine, it must be of the form X= 2*p_i *p_j *p_k where the prime factors are all distinct and all are congruent with 1mod4; an additional requirement is that X must be congruent with 2mod3. These criteria are necessary for X to be divine, but not sufficient.

In order for a number to be 8-fold divine it must have 4 odd prime factors that are distinct and congruent with 1mod4:
X= 2*p_i *p_j *p_k *p_l and X congruent with 2mod3. Here again, these criteria are necessary but not sufficient.

An example of an 8-fold divine number is 81770, the smallest 8-fold divine number I have found. All of its (a,b) values are prime.

81770 = 2*5*13*17*37
= 41² + 283²
= 53² + 281²
= 71² + 277²
= 97² + 269²
= 137² + 251²
= 157² + 239²
= 179² + 223²
= 193² + 211²

To date I have found two other numbers that have 8 solutions for (a,b) with 100% prime numbers as solutions: 896,090 = 2*5*13*61*113 and
1,496,690 = 2*5*13*29*397

This is what I see ...

Quote: teliot

This is what I see ...

link to original post

I'm seeing the same thing. Might be a formatting issue on the post because I don't have this issue on any others

I think I may have copied some cells from excel into word and then into the post. I am working to correct it, but so far no luck. I will try to get help from our IT guy.

In the meantime, I find that if you change the aspect ratio of the screen (by narrowing the window, if you are on a laptop or desktop) that the formatting problem goes away.

EDIT: I think I found the problem and fixed it. Within a spoiler box, I had a list of numbers that I was centering, but it was not closed correctly. When I fixed that, the problem seems to have gone away.

For those not familiar with "prime numbers congruent with 1mod4" they are presented here A002144

They are also called "primes of form 4*k + 1" and "pythagoran primes" and "rational primes that decompose in the field Q(sqrt(-1))"

Each of the pythagoran primes can themselves be written as a^2 + b^2. Ex: 5 = 1^2 + 2^2, 13 = 2^2 +3^2. Roughly half of all primes are congruent with 1mod4, the other half are congruent with 3mod4 (except for 2, which is 2mod4).

Being the product of pythagoran prime factors are the mathematical key to having multiple values of X= a^2+b^2.

Quote: gordonm888

I think I may have copied some cells from excel into word and then into the post. I am working to correct it, but so far no luck. I will try to get help from our IT guy.

In the meantime, I find that if you change the aspect ratio of the screen (by narrowing the window, if you are on a laptop or desktop) that the formatting problem goes away.

EDIT: I think I found the problem and fixed it. Within a spoiler box, I had a list of numbers that I was centering, but it was not closed correctly. When I fixed that, the problem seems to have gone away.
link to original post

Thanks, looks good now. BTW...interesting thread!

Cool find. Not sure if this thread is still being read, but some relevant sequences from the OEIS are

A088919 Smallest number having exactly n representations as sum of two squares of distinct primes.

and

A226562 Numbers which are the sum of two squared primes in exactly three ways (ignoring order).

(Not sure if new accounts can post links, but the sequence reference number is enough.)