Thursday, August 27, 2015

sequences and series - Uniqueness of real numbers represented as products of integer powers of primes




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 ekZ); 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=20305070110



227=21305271111




100=22305270110


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 peii1. Which is only possible if (ei) is eventually vanishing.


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