This morning’s gifts (which I could consider to be from the Queen of the Sciences Herself or from “Azarya’s Angels” who appear to serve as Her messengers to me), the mathematical insights it occurred to me in the early morning hours to ponder, relate to two concepts I have newly discovered: ceiling discrepancy and splitting roots. These insights and their descriptions will appear in my current paper in progress, “Telescoping the Integers,” just as soon as I have worked them out fully. In the meantime, I will explain them as best I can here.
In brief, with the benefit that description here will help me with description there:
A positive integer m has n positive divisors. The value of n relates to Euler’s Totient (or Phi) Function, and given the unique expression of m as the product of powers of primes, the number of ways to express m as a product ab of positive integers a and b is the number of distinct pairs of a and b, which correspond to every distinct possible product ab, which is n/2 unless m is a perfect square, in which case it is (n-1)/2.
Knowing m does not immediately (at present) give the number n. For sufficiently large m, whether m is prime or composite is a fact it may be difficult to establish with absolute certainty. When m is a known prime, or a positive integer of uncertain primality or factorization due to its size, there is one factorization that is certain and known: the trivial factorization 1 times m.
For any m, and for any known factorization into a pair of integers a and b, to include the trivial factorization, define a splitting root as the number c=(a+b)/2. I call c a splitting root because it splits m into the form (c+d)(c-d) for some d. In the case of m being prime or its factorization not fully known, the trivial factorization gives c=(m+1)/2 and d=(m-1)/2. This also gives an expression of m as the difference of two positive perfect squares. The positive difference between the square root of the greater of these perfect squares and the ceiling root of m (the square root of the smallest perfect square greater than m) is a quantity I will call the ceiling discrepancy of this factorization of m.
Given an m whose factorization or primality is unknown or uncertain, the study of the splitting roots and ceiling discrepancies which derive easily from the trivial factorization may be a topic for profitable further analysis of m.
Also it becomes plain to me that, in order to avoid dealing with fractional values of c and d, it is just as easy to divide out the factor of 2; i.e., analyses of these relationships can simplify matters by confining itself to the odd positive integers. Since these are as evenly distributed as the positive integers themselves, there is no loss of ease in looking for patterns and relationships.