Category: My Math Studies

Integer Factorization: Math Graphs du Jour
Advertisements These are graphs of the behavior of the terms of the Series of Ceiling Roots for m=11882081 starting with the basic Ceiling Root of 3449 and increasing by 2. (See past posts for explanation and discussion of Ceiling Roots.) This is the crazy sort of pattern of numbers I hope to tame and characterize, […]

Integer Factorization: Today’s Question
Advertisements Consider an odd positive integer m for which one is seeking factorization. Consider the more or less typical case where the Ascent of the Ceiling Square of m is large, is turning out to be in the thousands or more; that is, a value of m for which the iterations of my Fermatbased factorization […]

Integer Factorization: A 1313 Postscript, and the Next Example
Advertisements I generated the Series of Ceiling Squares systematically for each Ascent associated with yesterday’s example of m=1313. I also assembled the Beehive Plot, but won’t include that here, except to say that when the hive’s slope increased linearly, the bee line was a nice parabola. Let’s do another one, slightly larger, where the factorization […]

Integer Factorization: Let’s work an example.
Advertisements Let m=1313. The square root of m is between 36 and 37. (m+1)/2 is odd, and so the ceiling root and square should be odd as well. c=37 already is. m is c squared minus 56, between 49 and 64. If we then let c ascend to 39, m is c squared minus 208, […]

Integer Factorization: 18 Jan 2023 Update
Advertisements I am currently examining my table of odd composite numbers that I have used for the 3D graph and the charts I shared a day or two ago, sorted according to remainder, and at the distribution and factors of the odd composite numbers for each such remainder. This was what led me to notice […]

Integer Factorization: A Ceiling Square Tidbit
Advertisements When one subtracts an odd positive integer m from its appropriately adjusted Ceiling Square c2, with the adjustment being made by adding 1 to ceil(sqrt(m)) if necessary to make it even or odd as (m+1)/2 is even or odd, respectively, the remainder r obtained by c2m (so that m=c2r) is always congruent to 0 […]

Integer Factorization: New Charts
Advertisements These charts are from my Numbers file for a recent study of Ceiling Squares, Remainders, and Ascents for the odd integers 1 through 999, with definitions of the quantities indicated as follows: If m is a positive nonprime integer: Let c be the integer ceiling of the square root of m. Adjust the value […]

Integer Factorization: Series of Ceiling Squares, Take 3 (or 4)
Advertisements I embarrassed myself twice this week, having to take down insufficiently debugged algorithms and Perl code for not testing thoroughly enough. However, I now have an algorithm I’ve once again tested, and will test some more (yes, opening the possibility that a bug will require me to correct/retract), and this one has a symmetric […]

Integer Factorization: Drop back and punt.
Advertisements I had to backtrack and delete it post a couple of days ago, sharing my system for calculating Series of Ceiling Squares. It was not that the program was coded improperly, but rather that the algorithm’s math did not get rid of the “infinite telescoping” of the series for some value or another. I […]

And now, a message from the odd positive integers, my grandson Keegan, and myself.
Advertisements Part I: I decided simply to graph the values of the (Adjusted) Ceiling Square and its remainder for all of the odd positive nonprime integers I’ve put in my 3D graph so far (1 through 441). The result jumped out at me with its nearsymmetry of dots, which appear in the graph below. But […]