[dmurray]

> It's not that each congruence gives you new information

So it's not that each congruence gives you N bits of information, and you want kN bits in total. It's more like each congruence has a 1/k chance of giving you the full kN bits.

But in some information theory sense those are the same! Or concretely, if you were testing a large quantity of numbers in parallel, you would get information from each congruence.

[arcastroe]

I suppose you're correct. Even if you find a trivial congruence, you do get some information. Mainly: "It's not that one!" :)

The same information as trying two bases that don't form a quadratic congruence at all

[thaumasiotes]

Well, if a+b and a-b are difficult to factor, finding more congruences might give you another pair x+y, x-y, and if that pair is nontrivial but also difficult to factor, you may be able to make progress by looking at x+y / a+b and x+y / a-b. As long as they're not integer multiples of each other, the combination of the two facts is more informative than either fact alone, in the sense that you're now trying to factor a smaller number.