Suppose (a,b) = (c,d) = (x,y) = 1. Now x/y = (ad + bc)/bd. Let p be a prime dividing both (ad+bc) and bd. WLOG, assume p | b. Then p | ad, so p | d. So either (ad + bc, bd) = 1, in which case x = ad+bc and y = bd, or (b,d) is not 1. If (b,d) = 1, then if y is prime, WLOG b=1 and d = y and ay + c = x. If x = 1, we can't solve this in positive integers. So in general we must allow (b,d) > 1. --ms Dan Hoey wrote:
This may be completely dumb, but is it always possible to solve x/y = a/b + c/d, where all algebraic quantities are positive integers, for all x and y?
How can we make this not completely dumb? Given that x,y are positive and relatively prime, can you give formulas for a,b,c,d that guarantee that all are positive and that (a,b)=(c,d)=1?
Hmm, still pretty dumb. Suppose we also require that a,b,c,d are four different positive integers? I've gotten b and d different, but I haven't managed all four.
In case that's still dumb, can all of a,b,c,d be made relatively prime in pairs?
Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun