Research report follows, the first after a bit of silence, by way of a current example.
Consider the integers m1=3317, m2=4061, m3=5461, and m4=6077. They all have adjusted ceiling squares – the least perfect square with oddness/evenness the same as one half of one more than their value that is greater than each of them – which differ from them by exactly 164:
m1 = 3317 = 592-164.
m2 = 4061 = 652-164.
m3 = 5461 = 752-164.
m4 = 6077 = 792-164.
They are, as is to be expected, all congruent to 1 modulo 4. The number of iterations to factor each of them using the algorithm I wrote to employ “Fermat factorization” is, respectively in ascending order, 6, 9, 6, and 2.
Since their remainders are all 164, their pairwise differences immediately factor as differences of squares as well. I chose m1 through m4 as discrete semiprimes. I am, as always, looking at numbers on the small scale, like this, and on a larger scale to uncover elusive patterns. 164 = 132-(32-22), but can this give any real clue?
I’ll keep looking, of course, sniffing at this tree until it proves fruitful or not.