NOTE I have completely reworded this because I made a complete hash of it the first time, it got worse as I added to it. I apologize to anyone who might have been confused, and hope that this will be seen as a better attempt. Also, I'm not a (proper) mathematician so my terminology can be a little unconventional in places.
I have been investigating the Goldbach Strong Conjecture and have come up against a problem for which I can't find any references.
My approach is to prove that for any number in any given range of numbers there will be at least one pair of primes equidistant from it. I am attempting to do this (and have obviously not succeeded) by looking at the longest possible sequence of composites either side of a given number, regardless of whether that sequence is practically possible or not within the given range.
For an even number, the odd composites and primes will be an odd +ve or -ve offset from it. E.g. 60 is 1 away from 61 and 3 away from 63 on one side while it is also 1 away from 59 and 3 away from 57 on the other.
An odd number will conversely be an even offset from an odd composite or a prime.
I treat the odd and even cases separately.
To give this some meaning, I will try to explain what the following images represent.
Image 1 shows 0 and +-1 (mod 3)
Image 2 shows 0 , +-1 and +-2 (mod 5),
Image 3 shows 0, +-1, +-2 and +-3 (mod 7)
Take the number 60, it is 0 (mod 3) and so it is represented by the frame of Image 1 that has a single red dot. That's because the set of odd numbers that are divisible by 3 either side of 60 can be represented by a single series of period 6 (it's 6 because the prime in question is 3, and we are only interested in odd composites).
Now take the number 62. It is 2 (mod 3) which is the same as -1 (mod 3). The odd numbers that are divisible by 3 either side of 62 cannot be represented by a single series of period 6, but can be represented by 2 such series out of phase by 1 with one another, as shown in the frame of Image 1 with 2 red dots.
The number 64 is 1 (mod 3) and will be affected by the same pair of series as 62, hence the +-1 used to denote the pair of red dots.
Similar logic follows for mod 5, except in this case we have either a single series of period 10, 2 series of period 10 that are out of phase by 1, or 2 series of period 10 that are out of phase by 2 (or 3 depending on how you look at it) as depicted by the 3 frames of Image 2.
Image 1
Image 2
Image 3
Image 4
Now for Image 4 and the reason behind my question.
I wrote a program to test the different permutations of the first n odd primes and find the longest possible sequences of composites either side of an odd or even number.
Image 4 is a visual representation of the result I got for even numbers for the first 3 odd primes. My program shows that the longest sequence occurs for even numbers that are +-1 (mod 3) and +-1 (mod 5) and +-3 (mod 7). This is shown in the first 3 rows of Image 4. Because this result is for even numbers, the red dots in the bottom row of Image 4 are showing the odd columns for which a red dot appears in at least one of the first 3 rows. The black dot is the result. This represents the first guaranteed Goldbach Pair (for a number below $11^{2}$) - well maybe that's not 100% obvious but I have established that it's true.
In case you are wondering if I've lost the plot, I will make some practical sense of the result shown.
We can use the chinese remainder theorem to find even number solutions to the modular equations represented in Image 4 (just for the sake of this example):
- x=+-1 (mod 3)
- x=+-1 (mod 5)
- x=+-3 (mod 7)
One such solution (chosen because it is within the range $7^{2}/2
Now testing for equidistant primes either side of 46 we find:
{45,47} 45 is composite (Image 4 - row 4 - 1st red dot) {43,49) 49 is composite (Image 4 - row 4 - 2nd red dot) {41,51} 51 is composite (Image 4 - row 4 - 3rd red dot) {39,53} 39 is composite (Image 4 - row 4 - 4th red dot) {37,55} 55 is composite (Image 4 - row 4 - 5th red dot) {35,57} both composite (Image 4 - row 4 - 6th red dot) {33,59} 33 is composite (Image 4 - row 4 - 7th red dot) {31,61} Bulls-eye, a Goldbach pair at 46 +-15 (Image 4 - row 4 - black dot).
This does not prove anything in itself, but it does I hope provide some meaning to Image 4.
I was able to compute that for the first 11 odd primes, the greatest position of the black square using odd columns is
3,9,15,33,75,87,105,165,213,255,339 Series 1
I also checked the even columns for the first 10 odd primes, and got
6,12,24,42,60,90,150,180,216,312 Series 2
NOTE - Image 4 is not appropriate for odd numbers because a different set of modular equations is required: x = +-1 (mod 3); x= +-2 (mod 5) ; x = +-1 (mod 7). An appropriate solution for which is 43 which is +-24 from the nearest pair of equidistant primes {19,67} as predicted by the third element in Series 2
In case anyone is interested, one set of modular equations associated with 339 (the last member I have for Series 1) are
x=+-1 (mod 3); +-2 (mod 5); +-2 (mod 7); +-5 (mod 11); +-5 (mod 13); +-2 (mod 17); +-5 (mod 19); +-4 (mod 23); +-11 (mod 29); +-14 (mod 31); +-18 (mod 37)
I could not find any results on OEIS despite trying all variations of Series 1 and Series 2 that I could think of.
My question is, does anyone have any useful references, a formula (he said, not expecting an answer because it would probably prove Goldbach's strong conjecture) or an extension to the sequence(s) Series 1 and Series 2?