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.

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