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?
Algebraic Representability of Prime Number Generators
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number-theory
prime-numbers
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1What do you mean by "algebraically representable"? – 2012-06-01
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0A reasonable version might be: there is a nonconstant bivariate polynomial $P(z,w)$ such that for some sequence $p_n$ of distinct primes, $P(n,p_n) = 0$. – 2012-06-01
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0I guess that wasn't precise. By 'algebraically representable' I meant: http://en.wikipedia.org/wiki/Algebraic_function – 2012-06-03
3 Answers
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P×p=24×x+1(p is prime no )(any prime no square when divded by 24 remainder is 1)
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1Could you elaborate? It is not clear to me what is meant here. – 2017-12-13
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0$$\pi(x)=\sum_{n\le x}\left|\operatorname{sgn}\prod_{k=2}^n\prod_{l=2}^n(n-kl)\right|$$ is a representation that can be linked fairly directly to the definition of primes. – 2012-06-01
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1@anon I'm not sure that can be called "algebraic." – 2012-06-01
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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 $p_n\sim n\log n.$