I am wondering if there is a good method for finding two positive rational numbers that differ in their decimal expansion after 18 decimal places (i.e. the first 17 decimals should be the same). I want the fractions to have minimal denominators, where "minimal" here can be defined as minimizing the sum of the denominators.
I initially tried starting with something like 1/19 or 1/29 and trying to find a second one from there, but all my attempts produced far too large number for the second fraction. I though I would check with there was someone who had already thought through this and had a nice method already.