Involves mandelbrot.
Mandelbrot set= f(c) = c^2 + c
and you do it over and over again.
1: c
2: c^2 + c
3: c^3 + c^2 + c
4: c^4 + c^3 + c^2+ c
5: c^5 + c^4 + c^3 + c^2 + c
The pattern is obviously going to continue like that.
If we are writing this in terms for c=2, for 4 we would get
2^5 + 2^4 + 2^3 + 2^2 + 2 = 30
30 in binary (base 2) is 11110.
multiply it by 273 which in binary is 100010001
This turns it into 1111111111110 (8190) which almost looks like a mersenne prime. but is just 12 bits long.
BUT with this you can turn multiplying these big numbers (30, 273) into smaller numbers,
This one is as easy as 4 * 3. If you look at pic related you can tell the it is just the length of four 1's (30) getting laid out next to a copy of itself 3 times and the result is 12 bits. Its kind of indicitive of how it can be used to multiply numbers where the length of the bits is 4.
If we multiliplied 1111 (15) by 273, we would get 111111111111 (4095) which is the same 4*3 but a different multiplication result. You could theoretically do this for 1111000…00 for any amount of zeros times 273 and you would get a length of 12 bits.
If you use a different c at the top, like 10, you would get
1: 10
2: 100 + 10
3: 1000 + 100 + 10
4: 10000 + 1000 + 100 + 10
So 4 is 11110 but in tens. Similarly we could do similar multiplication but it would be 1111111111110 but in decimal, so same look of the number, different meaning.
You could do this for any base. So if we could find the 'correct base' this could be the way to shrink the number to be a mersenne prime type number and then factor the length of that number. I'm not exactly sure how. Or we could convert the number to a length in bits of a certain base then factor that number.