| I don't think that you will be able to get a neat closed form expression
for n.
I have no proof that this works for arbitrary values of e, b, i, and p,
but, being an engineer, I tried a dumb brute force experiment
in which I iteratively evaluated:
n' = 2 * ( e - b * i^n )/p/(i^n)
until | (n' - n)/n' | < .001
with initial value of n = 1 and a few sets of e, b, i, p that satisfied your
criteria.
It converged for most of the combinations I tried. It did seem to oscillate
for certain combinations. Empirically, this seemed to occur when
i got larger (> 1.05), but this is a wild leap based on a very small set of
observations. When this happened, I chose a value between the two oscillating
endpoints, and it converged. I didn't notice any divergence, but that doesn't
mean much.
Perhaps one of the numerical analysis wizards can help validate/invalidate
this kind of iterative solution or provide ranges and provisos. There may
well be unsafe ranges of non-convergence, but (practically) I would consider
using such an iterative approach until I found a combination of parameters
that didn't work.
If you use this technique, I would advise fail-safe'ing the program by
putting in iteration count limits, and maybe some divergence limits to
avoid floating overflow, bad denominator values, negative n, etc. A clever
program would also look for 'bouncing' or divergence and kick the value of
n0 into previously uncharted intervals. For example, n' = (n' + n0)/2
where n' and n0 are the oscillation endpoints.
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