# Tag: integer factorization

• ## Integer Factorization, Determinism, and Time’s Arrow

Advertisements A positive integer m greater than 1 has a representation as the product of powers of distinct prime numbers that is unique except for the order in the representation of m’s prime factors. One can establish uniqueness by requiring that the prime factors, regardless of their exponents, appear in ascending numerical order. If m…

• ## Integer Factorization: The Two Currently Apparent Best Leads

Advertisements First off, here are references to two earlier articles I have posted on the subjects that follow, by way of explanation rather than going over it all again: Integer Factorization: The Adjusted Ceiling Square, and Newly Deriving an Old Characteristic Polynomial A Method of Construction and a Theorem for Integer Factorization I have a…

• ## Integer Factorization: Still At It …

Advertisements . . . but otherwise, little to report. I’m filling in spaces where values are needed for the various columns in my nearly-complete chart of all the discrete semiprimes between 0 and 20000 whose Ascents are greater than 0; i.e., whose Ceiling Squares do not differ from them by exactly a perfect square, which…

• ## Integer Factorization: The Mystery Remains

Advertisements Exhibit A: The “Fivers” Yes, I am still barking up trees on this problem, even as particularly fruitful trees up which to bark become harder and harder to find. Behold! The quintessential devil in these matters! These are most or all of the odd positive discrete semiprimes less than 20,000 whose difference from their…

• ## Integer Factorization Update, June 14, 2023

Advertisements The values of f(x)=x2+2cx+r, for c being the Ceiling Root and r being the Remainder for an odd positive integer m we wish to factor, show patterns in their distribution that are useful. In particular, I have been looking at the function values for x=0,1,2,… in hexadecimal notation rather than decimal, and eliminating whole…

• ## Integer Factorization: An Important Graph

Advertisements Let odd positive m be the product of two unequal prime numbers p and q. Let the Ascent of m be the difference between (p+q)/2 and c, the Ceiling Root of m, which is the value of ceil(sqrt(m)) with 1 added, if necessary, to make c even or odd as (p+q)/2 is even or…

• ## A brief mathematical note

Advertisements I know better than to get too optimistic, but some of the numbers and relationships I am examining in my integer factorization study this week are seeming to show a possible way to compute smart, fast jumps to good candidates for factors using the Zone Grid. It’s not too easy a process to derive,…

• ## Integer Factorization: A Seemingly Vital Derivation

Advertisements For brevity, this post presupposes familiarity with material from earlier posts, namely this one and this one. I have just derived the precise quantities for the following: The line on the Zone Grid for a Remainder Series that represents odd positive integer m=c2-r has a linear equation with slope 2c-2(ceil(sqrt(r))-1. This formula makes this…

• ## Integer Factorization: The Ascent Is Just How Far Up You Go

Advertisements It occurred to me to articulate a simple relationship between odd composite integer m=ab, where a and b are odd positive integers, its Ceiling Square and Ceiling Root, and its Ascent, which is the least number n, always even, that one must add to the Ceiling Root c so that (c+n)2-m is a perfect…

• ## Integer Factorization: Bark, Bark, He’s Barking Again

Advertisements I am ironing out some new calculations that may come to nothing, but also may show me an advance into simplification. I was looking at the fact that a series of consecutive values of rn in any odd positive integer m’s Remainder Series will differ from a series of consecutive perfect squares by linearly-progressing…