Let pk represent the (k+1)th prime number. My hypothesis is that all positive real numbers may be represented as some infinite product
∞∏k=0pekk
(where ek∈Z); and moreover that this product is unique. Intuitively I am certain this is true, but I cannot imagine how I could go about proving it.
EDIT: some rational examples for clarity:
1=20⋅30⋅50⋅70⋅110⋅…
227=21⋅30⋅52⋅7−1⋅111⋅…
100=22⋅30⋅52⋅70⋅110⋅…
Answer
An infinite product ∏xi of real (and also complex) numbers converges only if the sequence (xi) converges to 1. In your case you have to have peii→1. Which is only possible if (ei) is eventually vanishing.
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