I decided to look at differences of squares near the values of numbers I wished to factor tonight. I took a particular randomly-generated semiprime m, and started generating differences of squares, sequentially, that would come out near m. The first graph I observed showed a triangle of fairly randomly distributed dots. But then I asked myself, “What happens to these dots if I make a line graph of just the y coordinate?” And then: “What happens if I take the second-difference function of that value?” For those of you coming in late – which is most of you – these are questions dealing with the quadratic polynomials I’ve been using to analyze integer factorization.
I then took the first and second difference of the sequence of y values I had generated, and saw that I had looped myself around, once again, to familiar territory. This graph is like many I have encountered showing this particular second-difference-values behavior. It’s pretty in its own way… but it brings me no closer to new discoveries.