Factoring[8 616 460 799] like 100 years ago

[Thomas Puettmann]

(09-08-2022 05:53 PM)Albert Chan Wrote:  Prime base has bad screening power.
Perhaps we should have pick base with high screening power? (low ratios)
(with this metric, base 22 is bad, pow-of-2, say 32, is good)

Yes and no.

The chains are not working independently if the numbers of links have common factors. My machine sieves in total modulo lcm(25, 36, 21, 26, 22, 29, 23, 31) = 18 627 909 300. If you substitute the 22 by 32 you sieve in lcm(25, 36, 21, 26, 32, 29, 23, 31) = 13 547 570 400.

In other words, the 32 excludes many times simultaneously to the 36, the 22 doesn't.

Another point is to save configuration time and material. This is why I was very happy with the 21, 22, 23, 25.

By the way, the numbers are not ordered to their size simply because then the bearing of the two counter wheels do not fit one besides the other.

But precisely the ideas you suggested lead me to the "optimal" didactic machine some time ago: Hand cranked with only two chains with 28 and 45 (or 35 and 36) links. When you factor e.g. 61577 with this machine, you have the first coincidence of two trace links at 82. A straightforward check reveals that 61577 + 82^2 = 68301 is not a square number. Thus, you continue cranking. The next coincidence happens at 152. And 61577 + 152^2 = 291^2. Hence, 61577 = (291-152)(291+152).