The following algorithm can be applied to a list of fractions and an integer input. You go down the list and multiply the input by the first fraction that will result in an integer. Taking this product as the new input, you repeat, using the same list of fractions. The algorithm ends when none of the fractions will result in an integer.

For example, if the list of fractions is {5/6, 5/2, 5/3}, then inputting an integer 2^a * 3^b will result in 5^max(a,b). A more complicated example: inputting 2^a * 3^b into {7/11, 11/(3*7), 1/7, (5*7)/2, 3/5} will result in 3^a (if a>0).

Find a list of fractions such that inputting an integer 2^a * 3^b will result in 5^ab.

How about a list that, when the input is 2^a * 3^b, results in 5^{a^b} (with b>0)?

The first list is ((13*2)/11, 11/(13*7), 1/13, (7*5)/2, 11/(7*3), 1/7)

The algorithm is

(A)Convert 11 into 13 and 2

(B)Convert 13 and 7 into 11

(C)Convert 13 into nothing 

(D)Convert 2 into 7 and 5

(E) Convert 7 and 3 into 11,

(F) Convert 7 into nothing

The idea is to first convert all 2 into an equal number of 7 and 5. Then, when you run out of 2, turn a 7 and 3 into an 11 so you can regenerate the 2, using the 7 which kept track of the number of 2. The process uses up a 3 and gives back an extra 13, which is removed once all the 7 are used up. Once all the 3 are used up, the algorithm ends after removing all 7.

EDIT: I messed up two of the fractions based on my algorithm, but it actually isn't right. I will post a tweak in another post.

Edited on November 22, 2006, 12:51 am

Posted by Gamer on 2006-11-21 22:42:31