Graeffe's root squaring method

[Albert Chan]

11-23-2022, 11:17 AM

If we don't mind negative coefficients, we can assume prime gap don't grow, and do gap of gap.
Maximum size of coefficients will drop by 2, or more.

XCas> p := makelist(ithprime,124+1,1,-1)      → [691,683,677,673,661,659,...]
XCas> p -= p[1:]      → [8,6,4,12,2,6,6,4,2,10,12,...]
XCas> p -= p[1:]      → [2,2,-8,10,-4,0,2,2,-8,-2,10,...]
XCas> min(abs(proot(p)))      → 0.806513599261


Assumption is not as good though (last prime gap we picked may be huge)
Above example, assumption is very good.

p125-p124 = 691-683 = 8 (which, we assumed the same for all future prime gaps)
p126-p125 = 701-691 = 10
p127-p126 = 709-701 = 8
p128-p127 = 719-709 = 10
p129-p128 = 727-719 = 8
...