Except when the prime factors of a, b already co-operate, the iron hand of the Fundamental Theorem of Arithmetic dictates that the integer powers a**pa and b**pb can never be equal for any two positive integer exponents pa and pb. However, in the jovial spirit of “close enough for government work”, we define two such powers to “hit” if their difference abs(a**pa-b**pb) multiplied by the tolerance is at most equal to the smaller of those powers. (This definition intentionally avoids division to keep it both fast and accurate for arbitrarily large integers.) For example, tolerance=100 expects a**pa and b**pb to be within 1 %.

For given positive integers a, b return the smallest positive integer exponents (pa, pb) that satisfy the tolerance requirement.

Input: Three integers (int).

Output: Tuple (or list) of two integers (int).

Examples:

assert list(hitting_powers(9, 10, 5)) == [1, 1]
assert list(hitting_powers(2, 4, 100)) == [2, 1]
assert list(hitting_powers(2, 7, 100)) == [73, 26]
assert list(hitting_powers(3, 6, 100)) == [137, 84]

The mission was taken from Python CCPS 109. It is taught for Ryerson Chang School of Continuing Education by Ilkka Kokkarinen