Tag: number theory
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Integer Factorization: Setting It All Down, Once Again
Advertisements Let m be an odd positive integer whose factorization we wish to determine. Each factorization of m into positive odd integer factors a and b, so that a<=b and m=ab, corresponds to an expression of m as the difference between two perfect squares, m=s2-t2, by setting s=(a+b)/2 and t=(b-a)/2. s will be even in…
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Integer (Fermat) Factorization Treated as Bodies in Linear Motion
Advertisements A chart, in search of a formula. Or, as I put it fancifully (with apologies to the advertisers for the late Chiffon Margarine and to Paramount Studios), “If you think it’s Stellar Cartography, but it’s not … it’s integer factorization!”
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Integer Factorization: New Study Year, New Strategy
Advertisements A deliberate approach is congealing in my mind for exploring integer factorization via the Ascent Chains I have mentioned lately. It has an advantage of being systematic and thus easily algorithmic, but I will see whether it is fruitful. The core of the technique will be to find the first ten or so small…
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Integer Factorization Anniversary: Back in the Saddle Again (Again)
Advertisements By Way of Review: Let m=c2-r be an expression of an odd positive integer m as the difference between the least perfect square c2 that is greater than m and is even or odd as (m+1)/2 is even or odd, and the remainder r one must subtract from c2 to obtain m. c2 and…
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Integer Factorization: During a Pause in Inspiration, Review
Advertisements These are the equations, probably previously posted in some form earlier on this blog, that relate the Zone Boundaries on the Zone Grid to the Characteristic Function corresponding to an odd integer m=c2-r with Ceiling Square c2 and Remainder r, which may give exploitable relationships to help in factorization of composite m.
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An Infinite “Heavenly” Hierarchy: Prime Numbers in 3D
Advertisements This scatter plot, and the idea behind it, owes its inspiration once again to Rebecca Newberger Goldstein’s “36 Arguments for the Existence of God: A Work of Fiction,” in which 6-year-old Hasidic Rabbi’s Son Azarya, a mathematical prodigy of amazing talent, discovers on his own the prime numbers, and ascertains them to be Angels.…
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A Curve of Ascension?
Advertisements In “36 Arguments for the Existence of God: A Work of Fiction,” Rebecca Newberger Goldstein’s Azarya, a young ultra-Orthodox rabbi’s son, sees the Prime Numbers as Angels. When I express prime numbers <= 6823 as c2-(s2-t2) for a strictly determined c, s, t, I get a curve rising heavenward for the scatter plot of…
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The World’s Oldest Construction Kit
Advertisements Well, sort of. I’ve been assigning each of the odd prime numbers, starting with three, a three-dimensional point in space, and decided to take a look at x-vs.-y (blue) and x-vs.-z (green) scatter plots, and it’s shown a definite shape, pattern, and order, especially with y and z on logarithmic scales, and I just…
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Characteristic Hyperbolae of Discrete Semiprimes and Their Factors
Advertisements an informal mathematical report by J. Calvin Smith, B.A., Mathematics, Georgia College, 1979, and Retired Member, American Mathematics Society written on Friday, March 8, 2024 at Mountain River Chalet, Talking Rock, GA, USA ABSTRACT: When examining the process of factorization of a discrete semiprime, it may be useful to develop and look at an…
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Integer Factorization: What I Have Shared With AMS
Advertisements Two weeks or so ago, I sent some important revelations I had had about integer factorization to the American Mathematical Society’s Review department. I did this, not because I knew the ideas I had formulated in my two-year study to be of great import, but because they could be. Now, in retrospect, I am…