Proving the Formula for the n-th Prime Number using Number Theory

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The discussion focuses on proving the formula for the n-th prime number, specifically p_n = ∑_{k=2}^{2^{n}}[ |n/(1+ π(k))|^{1/n}]. The user expresses confusion about the formula, mistakenly identifying the third prime as 2 instead of 3. Another participant points out an error in the formula, noting that it should use 2^n in the summation. There is a consensus that the formula needs further verification and correction. The conversation highlights the complexities of number theory and the challenges in accurately identifying prime numbers using formulas.
eljose
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Hello i need help to prove the formula:

p_n =\sum_{k=2}^{2^{n}}[ |n/(1+ \pi (k))|^{1/n}]

where []=floor function and |x| the modulus of x...this appear in number theory to evaluate the n-th prime however i am not able to prove it...:frown: :frown: :frown:
 
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So according to this formula, with n= 3, 2 is the third prime? I did not know that!
 
HallsofIvy said:
So according to this formula, with n= 3, 2 is the third prime? I did not know that!

:smile: Pure comedy!:smile:
 
Though in fact, that's 2^n at the top of the summation. Even then, something's wrong, I get the third prime as being three according to the summation. Need to check again since I rushed thru it, but it doesn't look right.
 

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