Beyond Binary Claims: Mathematical Discovery Through Human-AI Collaboration Part 7

Looking at Twin Primes

In the previous episode of this series, I mentioned that I was starting to look at π-complete numbers between twin primes. That has become even more interesting as I am starting to see some structure related to Order 2 π-complete numbers, those having the primes 2 and 3, and only those, in the factorization. Such numbers can be represented as

\((2^{e_2} \times 3^{e_3}) \\ \text{ where } e_2 \text{ and } e_3 \text{ are positive integers.}\)

These numbers are of interest in the twin prime question because all numbers bracketed by twin primes are divisible by 2 and 3 (therefore 6) because except for the special first case where 4 is between 3 and 5, the bracketed number has to be the one out of the three consecutive numbers divisible 3 (one of them has to be, and it can’t be the primes), and of course, every prime after 2 is odd, so the bracketed number has to be even (divisible by 2). I am calling the subset of Order 2 π-complete numbers that are bracketed by twin primes, “TPπ2” numbers. Here are some examples:

\((2^1 \times 3^1) = 6 \text{, or } \ (2^2 \times 3^1) = 12 \text{, or } \ (2^1 \times 3^2) = 18 \text{, or } \ (2^3 \times 3^2) = 72 \)

\((2^2 \times 3^3) = 108 \text{, or } \ (2^6 \times 3^1) = 192 \text{, or } \ (2^4 \times 3^3) = 432 \)

\((2^{148} \times 3^{27}) = 2,720,904,103,988,016,573,375,946,787,147,774,807,229,290,304,148,416,233,472\)

Notice, that except for the first case where both exponents are 1, the exponents will have the property that if one is even, the other will be odd. This carries through all future examples and I call this the “TPπ2 Exponent Parity Theorem” and have a proof.

The proof goes: both exponents cannot be even because that would make the number a perfect square, so it minus 1 is the difference of squares, so is factorable and not prime. If both exponents are odd, then the number has to have a last digit that is in the possible products of the last digits of the odd powers of 2 and 3. That turns out to be either 4 or 6. If it is a 4 then 4+1 = 5 and that side will be divisible by 5 and not prime. If it is a 6, then 6-1 = 5 so that side will be divisible by 5. The special initial case where both exponents are 1 works because 2 x 3 = 6 and 6-1 = 5, but 5 is the only number divisible by 5 that is prime. QED

The largest I have looked at so far is:

\((2^{6981} \times 3^{560})\)

which is 2,368 digits long. Also, the exponent of 2 is often larger than the exponent of 3, but not always, and they trade off the lead as the product increases in value. For example:

\((2^{49} \times 3^{320})\)

What draws my attention to this is that these TPπ2 numbers keep going and getting very big very quickly. That they get very big so fast means that they are very sparse, more sparse than general twin primes which are more sparse than ordinary prime numbers. I conjecture that TPπ2 numbers are an unbounded set, i.e. go on forever. If I can prove that, (beyond the empirical computational evidence I now have) it is a big discovery, because these are a subset of twin primes, so if they go on unbounded, it would prove The Twin Prime Conjecture.