Jarek found too many small & large solutions, I will just copy
one small and his largest:
16740062 / 824427647 =
8 + 9 = 17 < 18
539 + 543 = 1082 < 1084
> a) can you explain your method?
I am taking the number I want to approximate, e.g.
Then I expand it as continued fraction, which gives me good rational
approximations of the given number....how continued fractions give
rational approximations is a standard stuff...
After I get a good approximation, I have to check whether the
has less digits than used in primes generated by decimal extension,
required by the puzzle formulation.
> b) can you control the production of the integers p/q in the
> order to get prime numbers at least for one of them?
I have no influence on the p/q I produce. Perhaps I could produce a
more of the p/q's, but not by much. From the number of p/q's that
being produced, I feel that there should be examples with one of
prime, but very very rare. It is uncertain whether any such example
be found. It is highly unlikely both p and q can be prime at the
same time. I will let you know if I can find an example with prime p
Searching the files I had mailed you I have found that the
199155200426057675549 / 9808150846458521632654 =
21 + 22 = 43 < 44
has prime numerator p and the fraction
I wrote a little program using
continued fractions to get the best rational approximation.
There seem to be many solutions for which the number of digits
in the fraction is equal to the number of decimals but very few
for which the number of digits is less.
The next example higher than the one
where #digits is 31 and #decimals is