# Prime numbers function

1. Aug 10, 2004

### Anzas

is it true that this function:
f(n) = 3^(n)+2

will give a prime number for any natural value of n?

2. Aug 10, 2004

### Muzza

Nope, f(5) = 3^5 + 2 = 245 = 5 * 7^2.

Exercise: prove that f(n) assumes an infinite number of composite values.

Last edited: Aug 10, 2004
3. Aug 10, 2004

### mathman

To the best of my knowledge, there is no known algebraic expression that generates primes.

4. Aug 10, 2004

### Muzza

Well, you can get kind of close ;) http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html,

5. Aug 11, 2004

### Anzas

f(n) = 3^(2n)+2

where n is a natural number

6. Aug 11, 2004

### Muzza

No. Have you even tried looking for a counterexample? One exists in the really small natural numbers.

Last edited: Aug 11, 2004
7. Sep 18, 2004

### mathwonk

in the examples cited from wolfram it sounds as if one may have no clue which inputs actually give primes (i.e. positive outputs) and which do not.

8. Sep 19, 2004

### matt grime

It may sound that way since it is true.

9. Sep 19, 2004

### Gokul43201

Staff Emeritus
You can keep trying but you won't find a prime number function this way.

I think the only known single-parameter function that generates primes is the one involving Mill's constant : f(n) = [M^3^n]

10. Sep 19, 2004

### mathwonk

what is mills constant? the 3^n th root of 3?

this does not sound promising Gokul. unless this "constant" is like my brother the engineers "fudge factor", i.e. the ratio between my answer and the right answer.

actually isn't it obvious no formula of this type, taking higher powers of the same thing, can ever give more than one prime?

or are you using brackets to mean something like the next smaller integer? even then I am highly skeptical. of course the rime number graph is convex, so has some sort of shape like an exponential, by the rpime number theorem, i guess, but what can you get out of that?

maybe asymptotically you might say something about a large number, unlikely even infinitely many, primes.

but i am a total novice here.

Last edited: Sep 19, 2004
11. Sep 19, 2004

### CTS

12. Sep 19, 2004

### mathwonk

oh great, so "mills constant" is not even known. so the formula [M^(3^n)]. is not actually an explicit formula at all.

in fact apparently mills constant is computed by computing the primes instead.