Does this polynomial produce only prime numbers?

In summary, the conversation discusses the search for a general formula for an algebraic equation involving the behavior of a specific function. The speaker has found a general formula if a certain condition is met, but is seeking help in finding a counter-example. They are also asking if there is a polynomial of any degree that produces only prime numbers. A counter-example was given for n=7 and it was determined that no polynomial exists that only generates primes.
  • #1
AdrianZ
319
0
I'm trying to find a general formula for an algebraic equation, I'm studying the behavior of [itex]∏_{i=2}^n(1-\frac{1}{i^m})[/itex] for m=3 and so far I've seen that I can find a general formula if n^2 + n + 1 produces only prime numbers. if not, it would get way harder to find a general formula for it by my intuitive method. Does anyone have any ideas? Do you know a counter-example? I've tried n=1,2,3,4,5,6 and it seems fine. I think a counter-example wouldn't be so easy to find. anyone can help?
For the experts on this forum who are into number theory, is there a polynomial of any degree that produces only prime numbers?
 
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  • #2
It doesn't work for 7:

[tex]7^2+7+1=57=3\cdot 19[/tex]

I don't think a polynomial exists which only generates primes.
 
  • #3

1. Is it possible for a polynomial to produce only prime numbers?

Yes, it is possible for a polynomial to produce only prime numbers. However, it is not a common occurrence and would require specific coefficients and variable values.

2. How can I determine if a polynomial will produce only prime numbers?

There is no foolproof method to determine if a polynomial will produce only prime numbers. However, one method is to plug in various values for the variable and check if the resulting output is a prime number.

3. Are there any specific types of polynomials that are more likely to produce only prime numbers?

There is no specific type of polynomial that is more likely to produce only prime numbers. It ultimately depends on the specific coefficients and variable values used in the polynomial.

4. Can a polynomial produce infinitely many prime numbers?

It is unlikely that a polynomial can produce infinitely many prime numbers. Prime numbers are finite and rare, so it would be highly improbable for a polynomial to continuously produce only prime numbers.

5. Are there any real-world applications for polynomials that produce only prime numbers?

There are no known real-world applications for polynomials that produce only prime numbers. However, the study of these types of polynomials can lead to a better understanding of the distribution and patterns of prime numbers.

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