# Diophantine Equations and Fraction Sums

Considering the fraction addition problem a/c+b/d=(au+bv)/(LCM(c,d)), if the sum reduces then it is by a factor of the GCD(c,d). So, au+bv=n, where n is a multiple of a factor of the GCD.

The theory of Diophantine equations shows that there are solutions to this equation since u and v are relatively prime. Solutions exist for any n.

Given an initial solution, a_0 and b_0, using the Extended Euclidean Algorithm for n=1, general solutions are

a=a_0*n+vt

b=b_0*n-ut

for any integer t.

Since we need a and b to be whole numbers, i.e., integers greater than zero, we get the following inequality:

(-a_0*n)/v < t < (b_0*n)/u

Some of these values of a and b are not relatively prime to c and d respectively, which is a problem. However, this does give us another way to generate reducible fraction sums.

If we are concerned with all linear combinations of u and v, and not just those where the fractions a/c and b/d are reduced, then this could prove to be useful in finding the proportions of fraction sums that are reducible

Here is the start of a spreadsheet exploring this idea. It needs to be programmed.