# Aren't There Any Formulas for Prime Numbers?

• B
bagasme
Hello all,

We know that following formulas failed to produced all prime numbers for any given whole number ##n##:

1. ##f(n) = n^2 - n - 41##, failed for ##n = 41~(f = 1681)##
2. ##g(n) = 2^(2^n) + 1##, failed for ##n = 5~(g = 4,294,967,297)##
3. ##m(n) = 2^n - 1##, failed for ##n = 67~(m = 147,573,952,588,676,412,927)##
The question is: Are there any formulas that produce prime numbers for any given ##n## (without non-prime results), or aren't they?

Bagas

Homework Helper
Gold Member
2022 Award
Hello all,

We know that following formulas failed to produced all prime numbers for any given whole number ##n##:

1. ##f(n) = n^2 - n - 41##, failed for ##n = 41~(f = 1681)##
2. ##g(n) = 2^(2^n) + 1##, failed for ##n = 5~(g = 4,294,967,297)##
3. ##m(n) = 2^n - 1##, failed for ##n = 67~(m = 147,573,952,588,676,412,927)##
The question is: Are there any formulas that produce prime numbers for any given ##n## (without non-prime results), or aren't they?

Bagas

https://en.wikipedia.org/wiki/Formula_for_primes

• Klystron, kith and fresh_42
bagasme
Hmm,

I see the floor function recursivemethod interested.

In that case, why ##f_1## need to be irrational?

Homework Helper
Gold Member
I see the floor function recursivemethod interested.
I'm not sure what this means.
In that case, why ##f_1## need to be irrational?
Look at the denominators of the terms of the series expansion. What do you see?

Also, look again at the derivation of ##f_1##. Do you think this is really an interesting method of generating primes, or just a method of encoding primes that are already known?

I provide a new constant ##P_{buk} = 0.203005000700011...##. Derivation of the related prime generating formula is left to the reader.

• jbriggs444
Mentor
There is no useful formula for primes known, i.e. a formula that could be used to find new prime numbers more efficiently than clever trial and error.

Last edited:
• Janosh89
Homework Helper
Gold Member
Did you mean to include the word "known" in that sentence?

Mentor
It's always the current status of knowledge, a future proof of a nonexistence of such a formula would be really remarkable, but I added it explicitly for clarity.

Gold Member
Ignoring computational complexity for the sake of discussion, use of various 'sieves' seems to tell us that the interval between successive prime numbers should increase as we employ successively larger divisors. Then we encounter twin primes where the next higher odd number is also prime. Primes have fascinated me since I learned multiplication.

As pbuk and other posters have described, we do not so much generate prime numbers as discover them; like a miner finding a gold nugget in the next pan of gravel. As a student I attempted to compare intervals among prime numbers to square numbers, triangle, Catalan, and other sequences with instructive but inconclusive results.