JCSBimp's Hexagonal Orthogonal

“It’s amazing what a relatively lackluster Junior and Senior year in college math, followed by 33 years of working for the government bureaucracy, will do to one’s expectations of oneself in one’s major field.

“I am actually startled, especially in retrospect, that I’m sitting here now, at age 65, working out new results, and having to tell myself, ‘Calm down … calm down …’”

(from my Facebook post this morning)

I have discovered something I cannot quite yet fully characterize about a relationship between what I call the characteristic function of an odd positive integer’s representation in terms of its Ceiling Square, and its Remainder Line on the Zone Grid I’ve developed to study the integer’s remainders.

By way of recapitulation:

Let m be an odd positive integer. Let c² be the smallest perfect square greater than m that is even or odd as (m+1)/2 is even or odd. c² thus derived is the Ceiling Square of m, and c is its Ceiling Root. If m is not a perfect square, the remainder, a positive integer r=c²-m, is the first in a sequence of remainders r_n=(c+n)²-m. This remainder sequence is also the sequence of values of a quadratic polynomial in n: n²+2cn+r, for c and r defined as above, for n=0,1,2… . I have frequently referred to this as the Remainder Series of m.

Let the Ascent of m be the index (n) of the least positive integer in the Remainder Series that is a perfect square. This n is the number that one must add to c so that (c+n)²-m is also a perfect square, permitting “Fermat factorization” of m by the formula (s²-t²)=(s+t)(s-t).

I have developed what I call a Zone Grid by placing copies of the number line vertically on the nonnegative integer lines for x by the function assigning the point (x,y) to the integer x²+x+y. I realized that such a Zone Grid would allow me to look at values of the Remainder Series of m as falling along a straight line by locating the series as being values of the Zone Grid corresponding to successive vertical lines. “Zone Grid” refers to the fact that the perfect squares in this numerical arrangement all fall along straight lines, a family of them easily characterized by a straightforward formula. I am currently looking at these straight lines and where the Remainder Line corresponding to the Remainder Series crosses them. The first positive point at which the Remainder Line, starting at r₀‘s point and moving upward and rightward, intersects a Zone Line at an integer x value occurs when the value of x equals the Ascent. It is my intention to search for ways to use this result to develop new insights into integer factorization.

And now that you’re caught up…

Today I discovered what might be a simple relationship, but one that is pretty thrilling, between the polynomial x²+2cx+r and the remainder line for m=c²-r established above: Let v be the integer ceiling of the square root of r=r₀. Let y=ax+b be the remainder line. It turns out that (x-v)²+2c(x-v)+r will always be the remander ax+b added to the polynomial x(x+1).

I’ll be looking to see if this admittedly simple result leads to further insight.