- #1

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is it true that this function:

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

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

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

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

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- Thread starter Anzas
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- #1

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is it true that this function:

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

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

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

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

- #2

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

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

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

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- #3

mathman

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

- #4

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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

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how about the function

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

where n is a natural number

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

where n is a natural number

- #6

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

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- #7

mathwonk

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- #8

matt grime

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It may sound that way since it is true.

- #9

Gokul43201

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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.

- #10

mathwonk

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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.

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.

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- #11

- #12

mathwonk

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

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