At 05:17 AM 9/8/03, Jon Perry wrote:
Can someone characterize the integer solutions to:
ad + bc + bd = ac
These correspond to pairs of complex numbers (a+bi)(c+di) such that the real part (ac-bd) is equal to the imaginary part (ad+bc).
You've nearly answered your own question: if the product has its real and imaginary parts equal, it must be 1+i times a real. So, given any a+bi, the corresponding c+di must be a real multiple of 1+i times the conjugate of a+bi, (1 + i)(a - bi) = a+b + (a-b)i. To generate all integer solutions, given any integers a,b, let g = gcd(a + b, a - b). Then the corresponding values of c and d are c = k(a + b)/g, d = k(a - b)/g, for any integer k. (If you want to restrict to positive integers, you need to have a > b > 0, k > 0.) -- Fred W. Helenius <fredh@ix.netcom.com>