Does anyone happen to have at hand a short proof that demonstrates that there do (or do not) exist one or more algebraically representable prime number generating functions?
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You clarified that you meant, "Is there an algebraic function which generates the primes", that is, "Is there a fixed expression using only addition, subtraction, multiplication, division, and (fixed) root extraction which generates the primes?" The answer is no, because algebraic expressions cannot have the required growth rate, lacking anything that grows logarithmically, while $p_n\sim n\log n.$