# Finite algebra yielding mostly primes?

#### Loren

Does N^3-(N-1)^3, or any other non-trivial finite algebraic equation, generate mostly prime numbers?

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#### Maschke

What does mostly mean? I'm pretty sure (but can't prove) that every polynomial outputs infinitely many primes and infinitely many composites. I assume we're limiting to polynomials with integer coefficients.

By "mostly" we could possibly mean that the asymptotic density of one or the other is higher. For example the even numbers have asymptotic density in the naturals of 1/2. It's a way of quantifying some infinite sets where cardinality doesn't work.

The problem is that the primes have asymptotic density of zero. As numbers get large, the primes get further and further spread out. There are infinitely many of them, but there are huge runs of composites between them.

Questions like this are actually very deep as I understand it. I couldn't find any references that directly addressed your question but this article was interesting anyway.

• Loren

#### Loren

Thank you for the knowledge, including the link. They may be the closest I will get to an answer. (By "mostly" I indeed mean a density greater than 1/2.)

So density might not be a test for the rate of finite polynomials generating primes or not, like dividing zero by zero. Is the answer different for transcendental polynomials, infinite or not, of irrational coefficients?

Here's another: does an asymptotic density of zero for primes mean that p(1)/1+p(2)/2+p(3)/3+p(4)/4+p(5)/5+...+p(N)/N=2/1+3/2+5/3+7/4+11/5+...+p(N)/N is finite, where N is an integer?

#### Maschke

Here's another: does an asymptotic density of zero for primes mean that p(1)/1+p(2)/2+p(3)/3+p(4)/4+p(5)/5+...+p(N)/N=2/1+3/2+5/3+7/4+11/5+...+p(N)/N is finite, where N is an integer?
I have no idea. That's a pretty good question.

#### skipjack

Forum Staff
. . . non-trivial finite algebraic equation
Did you mean "expression" rather than "equation", even if the expression isn't a polynomial, and did you mean non-constant rather than non-trivial?

#### romsek

Math Team
err wow...

$\dfrac{8}{7}\ln(n) \approx \dfrac{Prime(n)}{n}$

so it appears that the sequence isn't finite

• Loren

#### Loren

Did you mean "expression" rather than "equation", even if the expression isn't a polynomial, and did you mean non-constant rather than non-trivial?
Your generalization and specification are warranted. Yes, the "equation" is better understood as an expression, and "non-constant" is more appropriate for my intention.than "non-trivial."

#### Loren

err wow...

$\dfrac{8}{7}\ln(n) \approx \dfrac{Prime(n)}{n}$

so it appears that the sequence isn't finite
I'm surprised that I don't remember coming upon this sequence before. Does your conclusion arise directly from the Prime Number Theorem?

#### romsek

Math Team
I'm surprised that I don't remember coming upon this sequence before. Does your conclusion arise directly from the Prime Number Theorem?
No. It comes from generating both sequences in Mathematica and adjusting the constant until they sat on top of one another.

• Loren

Forum Staff