Some naive musings about RSA

Sometimes in an idle moment, I find myself wondering about possible ways to attack RSA. Spoiler alert, I haven't found one.
But in passing, I found myself thinking about this:
For the uninitiated, RSA is based on the fact that if p and q are large primes, then the product p*q is a large number that is computationally infeasible to factorise - which is to say, no one's found a way, though quantum computers offer a possibility.
Let m be p * q, and n be (p-1)(q-1). Then RSA further depends on the fact that for any integer a, co-prime with m, a^n modulo m is 1. Since 2 will be co-prime with m, that means that 2^n modulo m is also 1. We can write that equivalently as
2^n - 1 = k * m
where k is an unknown integer. Actually, there are an infinity of possible k, but we're interested in the lowest (ignoring the trivial case of k = 0).
If we knew k, then that would give us n, or a factor of n (n is never prime, it is at least divisible by 4). The method I came up with is to start with a variable v initialised to m. Then repeatedly look for the lowest zero bit in v, say bit i, and add m * 2^i to v. That will set that bit, but not disturb any bits lower than i. So, just do this repeatedly, until v is a continuous sequence of 1s.
But remember the spoiler alert. Outside of some very improbable cases, for realistic values of m used in RSA, this will take longer than the time remaining in the Universe to terminate.
I think it's reasonably obvious that if the algorithm terminates, it will yield a value that is not greater than n. So here's the question: Leaving aside concerns about the longevity of the Universe, can one be sure that it will ever terminate?
Sylvia.