This is kinda recreational math with a goal in mind of progressing further toward a proof of the twin prime conjecture.
Consider this:
We start with a random prime: $109$
$3*109=327$
$327 \equiv$
$2 \pmod 5$
$5 \pmod 7$
$8 \pmod {11}$
$2 \pmod {13}$
$4 \pmod {17}$
$4 \pmod {19}$
$\ldots$
Or... as a list of remainders when divided by primes smaller than $\sqrt{ 3*109^2}$:
$$\{2,5,8,2,4,4,5,8,17,31,40,26,45,9,32,22,59,43,35,11,78,60,36,24,18,6,0,101,73,65,53,49,29,25,13,1,160,154,148,146\}$$
We can now start multiplying by odds. As long as it's list of remainders contains no 2's or 4's, and the odd we choose isn't greater than 109, we can guarantee that $(3*109*{odd}-4, (3*109*{odd}-2)$ is a twin prime pair.
My Questions
- Can we guarantee (read: prove) that our remainder list (after multiplying by some odd less than $109$) will contain no 2's or 4's (and thus that we've found a twin prime)? Perhaps there's an earlier limit? A later one?
- Spoiler Alert Brute force tells us that the first twin prime we come to is (middle 2 digits of) $7859$. I've yet to find pick a prime and get a further % of the way through possible odd numbers. Why not? Can anyone come up with a prime that gets further through these odds before hitting a twin? (perhaps use Mathematica) .