potential new algorithm for factoring semiprimes that are 5 mod 6

[IamMusavaRibica]

Hello, I wonder if i may get math related help in this forum, first i'll give an example and then my actual question at the end.

Let's take 7991 = 61*131 (but let it be p*q, unknown factors)

We know that if p is the smaller factor then p < sqrt(7991) < q
sqrt(7991) = 89.39..., let's just take 89 - D as the bound (and use D=0 for now, you'll see why I included D here).
And define:
p = 6a + 1
q = 6b + 5
Because p < 89, it follows that 6a + 1 < 89
6a < 88
a < 15
therefore b > 15

(6a+1)(6b+5) = 7991
36ab + 30a + 6b = 7986
6ab + 5a + b = 1331
We can substitute a = 15-c, b = 15+d, because of the above inequalities
6(15-c)(15+d) + 5(15-c) + (15+d) = 1331
6cd + 95c - 91d = 109
Notice that c and d must be different parity (one even, another odd)
so let's take c = 2m, d = 2n+1 (it doesn't really matter)
6*2m*(2n+1) + 95*2m - 91(2n+1) = 109
24mn + 202m - 182n = 200
12mn + 101m - 91n = 100 (divide by 2)
The coefficients near m and n are: 101 and -91. Adding up the absolute values yields 192, which is exactly the sum of our prime factors and gives the system of equations p*q = 7991, p+q = 192
If that wouldn't have happened, then substitute once again, but this time m and n must be the same parity, so m = 2t + 1, n = 2u + 1 or m = 2t, n = 2u, both would work.

Notice that this will happen if we (by mistake or by trial and error) define the smaller factor be 5 mod 6, and the larger one 1 mod 6 (in this case both c and d above would be negative).

You can also try this with any other semiprime, here are some numbers: 3713, 8633, 3977, this will still work the same.

However, this works efficiently when the difference of the two factors is very small. Here is where I'm stuck now. Remember the D i mentioned above? For example, when I tried 10476749 (which is 367*28547), I managed to get it working with D = -1868, -1867, 2950 and 2951. After some more testing it looks like it actually depends on the difference between two prime factors. My question is, would there exist any way to determine D based on any factors? With certain numbers, it would be very slow to bruteforce.

[SethTro]

You might check out https://en.wikipedia.org/wiki/Fermat...ization_method which looks similar to what you are doing.