Correlation betweeen k binomial variables


This is my first post as a new member, so hope I'm posting this in the right place...

I need a general formula to calculate the minimum sample size N required to determine whether k binomial variables are somehow correlated (either positively or negatively) at significance level p<= alpha, where:

N = sample size
k = no. of binomial variables, from i= 1 to i=k
p(i) = probability that variable i is "positive" (otherwise negative)
alpha = maximum p-Value

For example, how many times must I toss a set of k biased coins, in order to determine whether any of them somehow talk to each other at significance level alpha?

In this case the Null Hypothesis is that the k coins act independently according to their respective probabilities p(i=1 to k), so I think we're looking at a 2-tailed distribution where a correlation could be either positive or negative. But I'm not sure.

Is there a general formula for the actual p-value in terms of N, k and p(i), which I could then solve/rearrange for N in terms of k, p(i) and alpha?

I'm really struggling with this, so any help would be appreciated, thanks!

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So I figured out the following formula:

α = [p^k + (1-p)^k]^N - [p^(k*N) + (1-p)^(k*N)]

Now I just need to solve (rearrange) this formula to find N in terms of p, k and α, but I'm struggling with the algebra...

Any help, please???
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