*Please see my wiki for a comprehensible version of this post.*

I was pretty pleased that I factored a recent Composite of the day entirely in the complex plane.

The number was 9509, which I noticed immediately is 97²+10². Since I know Fermat's little theorem, and I know that the Composite of the day is composite, I knew there should be another way to write it as the sum of two squares. A little bit of counting (100+193+191) showed that it is also equal to 95²+22².

For some reason, I know that if I use those two summations to write complex integers with modulus 9509, their greatest common factor will also divide 9509.

So I said, (97+10i) - (95+22i) = 2-12i. The modulus of that is 2²+12²=148. The factors of 2 must be irrelevant (since the Cotd is odd), so

**37**should be the number we're looking for.

Similarly, (97-10i) - (95+22i) = 2-32i. The modulus of that is 1048. Again, discarding factors of 2, we're left with the prime

**257**.

And the number is now factored, by finding complex integers with the right modulus and manuipulating them in the complex plane.

Pretty wild, huh?

Man, for years I thought this was called the little theorem, but apparently that's something else. The one I meant has no cool name: http://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares.

ReplyDeleteAlso, today's composite of the day is the difference between two fourth powers. That must be very rare.

ReplyDelete