Tag: number theory

My Integer Factorization Study: This Isn’t Finished.
Advertisements After my Tiny Eureka yesterday led to nothing new in the way of mathematical discoveries – led, in fact, to a tautology, a sort of uninformative a=a result – I took a breather. I had already scheduled myself a break after “Tiny Eureka,” and it felt strange to pull away from the study on…

My Integer Factorization Study: May 7 2022 4:29pm Eastern Update
Advertisements This morning’s Tiny Eureka demonstrates itself right now to be a trivial finding, and therefore unlikely to lead me to a Big Eureka.

My Integer Factorization Study: Tiny Eureka! (or Should I Say “Woof!”?)
Advertisements I am finally barking up the correct tree, so to speak, and I have found the little squirrel I sought. From the beginning, with gusto! (By the way, the relationship I am about to describe took a while to hammer out on paper, and a great deal longer to get right in Perl code.)…

My Integer Factorization Study: Discouraged, but Then Renewed in Hope
Advertisements UPDATE: the sometimes triangular tables of values I have been building over the last several days have borne fruit, so to speak. I do not know where this will lead me, or how much easier it will make the task of integer factorization, but I have found a relationship between the product m of…

My Integer Factorization Study: Closer, but No Cigar (Squirrel?) Yet
Advertisements Last night and this morning, I began looking at the behavior of the remainder r = s2 – m, where m is the product of distinct primes p and q, p < q, and s is the smallest integer greater than the square root of m. I began to build a kind of triangular…

My Integer Factorization Study: Boiling It Down
Advertisements It now seems that the task at hand is this: Given two linear equations with integer coefficients: y1 = a1x + b1, and y2 = a2x + b2, find the integer values of x, if any, where y1 is an integer multiple of y2. y1 = n*y2 when x = (b2n – b1)/(a1 –…

My Mathematical/Number Study: What Got Me Up Early This Morning
Advertisements Consider the positive integer m which is the product of distinct primes p and q. Consider s, the smallest integer greater than the square root of m, and the difference r = s^2 – m. p will be less than s, and q will be greater than s. Thus p = s – c1…

My Number Study: Possible Breakthrough?
Advertisements This is a continuation of progress reports on a number study of the integer factorization problem I have been conducting this Spring. Last night, in working through my calculations to see which quantities involved I could express in terms of other quantities, I came upon one that surprised me, and I wonder if the…

My Mathematical Study: A Brief Revelation/Realization
Advertisements When m is the product of two odd primes p and q, with p < q, and m is also equal to s^2 – r where s is the integer ceiling of the square root of m, the polynomial x^2 + n*x – (r + s*n) for the positive integer n that allows it…