I'm rapidly running out of time to spend on this. I'm sure you think you're being clear, but I really can't follow much of what you're saying. Let me show you

I'm rapidly running out of time to spend on this. I'm sure you think you're being clear, but I really can't follow much of what you're saying. Let me show you. > N=163123711. Right, so that's my example. > choose a number higher than N and has many different factor pairs, of which the difference of the factor pair is lower N. So basically you pick a number S > N which is smooth (has lots of small factors) Then you appear to find lots of solutions: (a+c)a = S Specifically you require c<N. Example: 79534 × 81345 = 6469693230 -> (79534+1811) x 79534 = 6469693230 So 1811 is a candidate for c. OK. > First we write down all the c's possible. So you write down every c such that there is an a with (a+c)a=S, and because of your choice of S there will be many such values. > After this we start the algorithm with our first number "6469693230": 6469693230 % 163123711 = 107868501 What? > Now because we know that 1x107868501 = 107868501, we know the c of the number we left with, 107868501-1 = 107868500. > c = 107868500 > Does 107868500 match the c's we had written down? What? > (a+c) x a - (b+c) x b > (79534+1811) x 79534 - (1+107868500) x 1; c is not the same (1811 is not 107868500). > (79534+2137) x 79534 - (1+107868500) x 1; c is not the same (2137 is not 107868500). What? You haven't actually explained your algorithm. I've got as far as: I have a collection of candidate values for c. You haven't explained then what you're actually doing. But now I really have run out of time. If you write this up properly somewhere and point at it I'll try to come back to it, but no promises.