As I’ve previously shared, my math study seems to lead me in kind of a circular path, between periods of great excitement at potentially profound discoveries, and “letdown” periods where I realize what I “discovered” is something not only already evident, but blindingly obvious.
I still want to remain happily agnostic about which of these poles of mood/emotion/thought is the more accurate one.
RECAP – September 21, 2002
Let m be an odd integer we want to factor. Let c be the integer ceiling of the square root of m, and let r be the difference between c2 and m.
I call c2 the Ceiling Square of m, and c the Ceiling Root of m.
Define a Characteristic Polynomial for m as follows: f(x)=x2+cx+r. The values of f(x) for non-negative integer values of x lead to all factorizations of m when a particular x leads to f(x) being a perfect square. If m is a prime number, f(x) will still have a perfect square value which corresponds to the factorization m=1*m. The values of x are those for which (c+x)2-m is a perfect square, yielding an expression of m that allows “Fermat factorization”; i.e., a difference of perfect squares yielding to the formula (s2-t2)=(s+t)(s-t). It is when s and t differ by 1 that one finds the “trivial factorization” of m=1*m.
Now, this is quite obvious after some calculation. One merely has to complete the square of the characteristic polynomial, remember that m=c2-r, and rearrange terms slightly, to get:
It hardly seems worth hammering at large amounts of both small and large discrete semiprimes to find, does it? And yet, in my mind it both simplifies the problem and leaves that tiny crack of an opening in the barrier to further discovery.
Yes, I know mathematics is dry, and coming back again and again to the same general neighborhood of realization and observation is even more dry. I suspect any mathematical experts following this blog, if there are, have either tired of the round-and-round or long since given up on my unearthing any actual profundity. And so I apologize to the long-suffering. You are being held by my hope that there really is something else to find here, and … to be honest, my not having found it yet is burdensome to even the most patient.
I thank you for being here. And I still hope I’ll have something for you.
Next: Characterization of the non-integer values of x which produce all perfect square values of f(x).
Maybe exploring this will lead to more than a simple letdown.