Beyond Binary Claims: Mathematical Discovery Through Human-AI Collaboration Part 9

Getting into Big Numbers

I am continuing to work on my twin prime subset proof. As I mentioned in Part 8, I can show that the twin primes that bracket TPπ2 numbers (those that can be expressed as 2ᵃx3ᵇ for some a and b > 0) go on without bound as the exponents of 2 and 3 race against each other to higher values. After the simple proof of the Exponent Parity Theorem (the exponents can’t be both even or both odd) I use the Hardy-Littlewood conjecture to show that continuing the race is always below the expected density of twin primes, and show that neither exponent can be forever less than the other.

If my proof that the set of TPπ2 numbers is unbounded holds up, it provides a means of attack for proving the general Twin Prime Conjecture. Some would argue that having found a subset of general twin primes to be unbounded, I have proved the conjecture, but strict number theorists will still require proof that given any TPπ2 number, there must be an ordinary twin prime that is larger. That will be the next goal.

Of course, I am also interested in proving my conjecture that all sets of TPπn numbers are unbounded. Also, I am looking at π-complete numbers of the form r#·m where r# is a primorial number (the product of the first r number of primes, not the product of prime numbers less than r) and m is an integer that has no prime factor greater than r (m is “smooth” with respect to r), that are bracketed by twin primes. Specifically, I am trying to find the smallest m for each r that places their product between twin primes.

As far as anyone knows 2# = 6, 3# = 30 and 5# = 2,310 are the only pure primorials (m = 1) that are between twin primes, as in: (5,7), (29,31) and (2309, 2311). I have found a few where m = 2. Those are: 1#·2 = 4, 30#·2 = (a 47 digit number), and 148#·2 = (an enormous number). I am calling these numbers that are the first after a primorial that are between twin primes, “First of Order” (FoO) numbers. Except for those first few, FoO numbers are all enormous. The values of m range from 2 to the largest I have found in 471#·435,888 which may seem large, but 471# is so huge that the multiplier is insignificant. Also it may look like 507#·712 is smaller because it has a small multiplier, but 471# looks tiny next to 507# because these numbers grow super exponentially. I conjecture that FoO numbers are unbounded.

For the TPπn where n > 2, I have not found special patterns of exponents (yet) that show structure the way it is there for TPπ2 numbers. However, I am looking at the distribution of these numbers as they appear with the ordinary twin primes. I have run programs to sift through the first half-trillion numbers, counting the twin primes and each oder of TPπn numbers. Here is that data:

Highest π-complete number bracketed by primes: 501,382,801,920
Total number of twin primes found:                 988,739,362
Counts of TPπn numbers with total:                       1,696
TPπ1:  1
TPπ2:  18
TPπ3:  88
TPπ4:  247
TPπ5:  360
TPπ6:  436
TPπ7:  362
TPπ8:  141
TPπ9:  43
TPπ10: 1
TPπ11: 0
TPπ12: 0

In general, π-complete numbers get more rare going up the number line. Of course, only a small fraction of those will have prime numbers on both sides. TPπ2 numbers are so rare that out of the 66 I have found (so far) only 18 are less than 500 billion, and yet, those are the ones that can be shown to go on, without end.