Prime numbers function

  • Thread starter Anzas
  • Start date
  • #1
87
0
is it true that this function:
f(n) = 3^(n)+2

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

Answers and Replies

  • #2
695
0
Nope, f(5) = 3^5 + 2 = 245 = 5 * 7^2.

Exercise: prove that f(n) assumes an infinite number of composite values.
 
Last edited:
  • #3
mathman
Science Advisor
7,984
513
To the best of my knowledge, there is no known algebraic expression that generates primes.
 
  • #4
695
0
Well, you can get kind of close ;) http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html,

However, there exists a polynomial in 10 variables with integer coefficients such that the set of primes equals the set of positive values of this polynomial obtained as the variables run through all nonnegative integers, although it is really a set of Diophantine equations in disguise (Ribenboim 1991). Jones, Sato, Wada, and Wiens have also found a polynomial of degree 25 in 26 variables whose positive values are exactly the prime numbers (Flannery and Flannery 2000, p. 51).
 
  • #5
87
0
how about the function
f(n) = 3^(2n)+2

where n is a natural number
 
  • #6
695
0
No. Have you even tried looking for a counterexample? One exists in the really small natural numbers.
 
Last edited:
  • #7
mathwonk
Science Advisor
Homework Helper
2020 Award
11,245
1,452
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
matt grime
Science Advisor
Homework Helper
9,420
4
It may sound that way since it is true.
 
  • #9
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
7,082
20
Anzas said:
how about the function
f(n) = 3^(2n)+2

where n is a natural number

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
mathwonk
Science Advisor
Homework Helper
2020 Award
11,245
1,452
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:
  • #12
mathwonk
Science Advisor
Homework Helper
2020 Award
11,245
1,452
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.
 

Related Threads on Prime numbers function

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
24
Views
6K
  • Last Post
2
Replies
28
Views
7K
  • Last Post
Replies
1
Views
2K
  • Last Post
2
Replies
44
Views
11K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
9
Views
3K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
2
Views
2K
Top