To give a mental spark to my mathematical searching, I am listing here some of the basic principles I have found or realized so far, none of which are revolutionary, but all of which I still hope will lead me to some revelation that is so.
- [the usual definitions] Let positive m be a discrete semiprime, i.e., the product of distinct prime integers p and q, ordered such that p<q: m=pq.
- Let c=ceil(sqrt(m)), or the integer ceiling of the square root of m. I call c2 the Ceiling Square of m, and c the Ceiling Root. Let r be the difference c2-m, so that m=c2-r.
- Because of the product of odd primes being itself odd, when c is even, r is odd, and vice versa.
- A result from Fermat gives that for each possible integer factorization of an odd integer (since m is a discrete semiprime, there is only 1 times m and p times q), it can be expressed as the difference of two perfect squares. Specifically, m expressed as a times b gives us m=s2-t2, where s=(a+b)/2 and t=(a-b)/2. Since squaring makes the sign of a-b of no consequence, one can take s and t to be their absolute values, and adjust the a-b term of t accordingly.
- When the factorization of m under consideration is the trivial 1 times m, this becomes s=(m+1)/2 and t=(m-1)/2, consecutive integers. Every odd positive integer has this s2-t2 expression. For a discrete semiprime, there is one other s and t pair to discover; for other odd composite numbers, there are more than one.
- For all the possible ways to express an odd composite integer as the difference of two perfect squares, the s term will always be odd and the t term will always be even, or the s term always even and the t term always odd, corresponding with whether (m+1)/2 is odd or even, respectively.
- However, it is readily evident that for a given discrete semiprime m, or indeed for any odd composite integer, whether its values of s are even or odd does not determine whether its Ceiling Square and Root are even or odd.
- When m=pq has Ceiling Root c, one can determine positive integer constants c1 and c2 so that m=pq=(c-c1)(c+c2)=c2+(c2-c1)c-c1c2. This gives an alternate expression of r, the difference between the Ceiling Square of m and m, previously given, as c1c2+(c1-c2)c.
- Since m is odd, c1 and c2 are either both odd or even.
- (c2-c1)/2 is a number I have called the Ascent of the factorization of m into pq and behaves as follows: It is the value of s-c, for s belonging to the pair of values of s and t whose s2-t2 representation of m factors m nontrivially. (The Ceiling Root c thus “Ascends” to the desired s.)
There. Now it’s all in front of me, and all in one place, fresh in my mind.
Now to keep looking for new relationships between c, r, s, and t! Legendre, Moebius, and Mr. Reciprocity, you all might be of some use here! But who can say?