>>4239

I think I found a way to generate primes. Unconfirmed but I hope you get your hopes up

>>4252

Look at the triangular numbers prime factorization. Look at the next one and the previous one. The triangular number will share some amount of factors with the next one, and the rest with the previous one. This pic is all triangular numbers up to like a thousand on the left (so pixel at depth 5 is 5(6)/2=15) then the horizontal axis is factors of that number. They are always in vertical pairs.

Start with 3

3 = (3)

6 = (2,3)

10=(2,5) (genrated 5)

15 = (3,5)

21 = (3,7) (generated 7)

28 = ((2,2),7)

36 = ((2,2),(3,3))

45 = ((3,3),5)

55 = (5,11) (generated 11)

66 = (2,3,11)

78 = (2,3,13) (generated 13)

etc.

So you can get generate primes through the triangular numbers.

QED??

>>4257

Also it looks A LOT like our grid but shifted the other way

>>4251

I screwed with these all day. I think he told us this because for even squares we can just look at them like 4 smaller squares. We can do this until we get smaller numbers.

145 = 12^2 + 1

= 4*6^2 + 1

=4^2*3^2 + 1

Then at this point we have a odd square which we may be able to do stuff with. Just trying to fit together the 1+8T and the divide the square by 2 crumbs.

>>4261

= 443*3 + 1

= (T(4) + T(3))(T(3) + T(2)) + 1

= T(4)T(3) + T(4)T(2) + T(3)T(3) + T(3)T(2) + 1

= 106 + 103 + 66 + 63 + 1

= 60 + 30 + 36 + 18 + 1

= 6(10 + 5 + 6 + 3) + 1

= 6( 3(5) + 3(3)) + 1

= 18(5+3) + 1

Something like this process for the factoring 145.

>>4263

So you're still going to need to factor, but no. Check this example out:

These are (x, T(x), factors of T(x))

(15, 120, ([3, 5], [2, 2, 2]))

(16, 136, ([2, 2, 2], [17]))

(17, 153, ([17], [3, 3]))

(18, 171, ([3, 3], [19]))

(19, 190, ([19], [2, 5]))

(20, 210, ([2, 5], [3, 7]))

(21, 231, ([3, 7], [11]))

So for example, T(17) you know the factors that match with the next are (3,3), so you take T(18)/(3*3) and you get 19. So you don't necessarily need it to be prime. You could just crank through this list

>>4265

The more I think about it the more this just intuitively follows from n(n+1)/2

>>4266

Even more, It seems to just generate every single number. This might be crap. :(

(15, 120, ([3, 5], [2, 2, 2])) (15,8)

(16, 136, ([2, 2, 2], [17])) (8,17)

(17, 153, ([17], [3, 3])) (17,9)

(18, 171, ([3, 3], [19])) (9,19)

(19, 190, ([19], [2, 5])) (19,10)

(20, 210, ([2, 5], [3, 7])) (10,21)

(21, 231, ([3, 7], [11])) (21,11)

^proof this is bunk