This is related to the the May 16 post, but takes only the prime indexed terms. Does it still diverge?
Hint
Transform the product into a sum
Hint
The harmonic series 1 + 1/2 + 1/3 + ... 1/n +... diverges
this post was submitted on 31 May 2024
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I don't know how to begin proving it, but the more I run this series out, bigger it gets. The conditions of the equation are such that it will always have a consistently non-zero rate of increase, even though that rate of increase decreases each time the formula is cycled ((p~n~/p~n~-1) will always be more than (p~n+1~/p~n+1~-1), nonetheless any and every (p~n~/p~n~-1) will be >1). The divergence will be glacial, but definite.
I can confirm that your intuition for divergence is correct.