Tag: number theory
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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…
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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,…
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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…
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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…
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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…
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Integer Factorization: An Overdue Update
Advertisements New revelations in my study of factorization, Ceiling Squares, and the Zone Grid have not been coming lately, but I have indeed been looking at the data and trying out new ways to visualize their relationships, every day, and my work is still underway, despite no evidence of these facts in this blog. Right…
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Integer Factorization: Update, Saturday 15 April 2023
Advertisements The most tantalizing current avenue in my study is to continue examining the relationships between m’s trivial factorization arising from its expression as ((m+1)/2)2-((m-1)/2)2, its remainder line on the Zone Grid, and the zone boundaries thereon. I am looking at 427 of the odd discrete semiprimes with the smallest prime factors in hopes of…
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Integer Factorization: This Morning’s Delivery
Advertisements It is a rainy Saturday of Holy Week, 2023, smack-dab between Good Friday and Easter/Resurrection Sunday on the religious calendar which I personally celebrate. It is also a year, give or take a week, since inspiration started coming to me in the early morning hours, usually after our chugs woke me up to have…
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Integer Factorization: The Great Trapezoid
Advertisements In an effort to elicit potentially useful patterns from my mathematical study, and particularly from the Zone Grid idea I discovered, it would seem that I have come up with another way to help me visualize what is going on with the Remainder Series associated with the Ceiling Square and its Ascents. This one…
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Integer Factorization: What’s Up Lately (2 April 2023)
Advertisements What’s got me all excited lately mathematically has been further exploration of the odd positive integers and the way their remainder series behave when translated into Remainder Lines on the Zone Grid. I established the construction of such a grid and the lines in question in a post a few weeks ago, and now…