Tag: number theory
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Integer Factorization: The Well-Ordered Forest . . .
Advertisements . . . still challenges me with the distribution of its trees! Ascent and Remainder and how they chain together series of numbers in various ways indeed shows itself to be a clarifying revelation to me. My previous post gave some of the clarification of the subject matter that occurred to me in the…
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Integer Factorization Via Remainder Analysis
Advertisements Preliminary Principles and Examples by J. Calvin Smith, Retired Member, AMS B.A., Mathematics, Georgia College, 1979 Thursday, February 8, 2024 SYNOPSIS: This blog post outlines preliminary steps to take in the factorization of an arbitrary odd positive integer m, using new concepts the author has defined and developed over the course of the past…
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Integer Factorization: An Elementary but Powerful Rule
Advertisements It only takes a little bit of working with polynomials having integer coefficients to realize an important result that has seen wide publication in mathematical literature: If f(x) is a polynomial in x with integer coefficients, and the prime p divides f(x) for a particular value of x, then p will also divide f(x+np)…
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Ascent Chaining: Degenerate Cases
Advertisements So far, in that part my study of integer factorization that has to do with “Ascent Chaining,” I have, in order to make the equations work right, liited my attention to those values of x2-r, for r not a perfect square, where x is greater than (r+4)/4, so that (x-1)2 < x2-r < x2…
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Integer Factorization: The Regularly Scheduled Pause to Reassess
Advertisements I said last night, to a math teacher who is in my choir, that the math study now feels like a trunk I’m trying to open that is sealed with a thousand padlocks. Once I’ve unlocked one, I’m seemingly no closer to opening it, and I’ve just got to wait for the next key…
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Integer Factorization: Some Ascent Chaining Examples
Advertisements This is a lengthy and more informative follow-up to last night’s post which gave some informal definitions and background to what I am about to discuss. You might need that background, since I’m mostly plunging right into things here. I verified in further research last night that Ascent Chaining does indeed show consistent and…
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Integer Factorization: Glimpses of a Pattern
Advertisements I am seeing beautiful patterns and connections between odd positive integers which share certain characteristics related to the concepts of Ceiling Square, Remainder, and Ascent, concepts I have developed over the last two years and have explored more or less feverishly in the months since I discovered them through my own study and inspiration.…
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Integer Factorization: High C and Low C
Advertisements So I’ve figured out that not too far above and below the Ceiling Square c2 of the representation of positive odd composite integer m as c2-r, there are other perfect squares x2 which, when you subtract r from them, produce a value that has a factor in common with m. Given that these new…
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Integer Factorization: Ascent/Remainder Chain Grids
Advertisements This is a brief post to inform readers that I am currently exploring two-dimensional arrangements of Ceiling Square and Remainder pairs and an ability I noticed to chain them together in two ways: chains of values of identical Ascent whose factors arrange to satisfy a quadratic polynomial, as do their Ceiling Squares; and chains…
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Integer Factorization: Something Interesting and Simple, and a New Perl Script
Advertisements In the course of investigating integers that I can chain together by using Ascent Chains, as well as divisibility rules for polynomials with integer coefficients (in particular, if such a polynomial f(x) evaluated at a particular has n as a divisor, then so does f(x+kn) for all integers k), something very simple occurred to…