Does there exist an f(x) minimum first-degree polynomial with integer coefficients, such that for any x integer number, f(x) is prime?
Any help would be appreciated!
No!
Proof by reductio ad impossibile.
Suppose that there is exist such polynomial .
Let now such that , when is prime.
Let now be , and let us look at:
is polynomial with integer coefficients.
So, , but is generates primes only, therefor for all integer .
Now, from the fact that polynomial can't have the same value more than times, we get the contradiction!
Here some interesting polynomials:
when then is prime...
Also(famous one):
There are many such functions. For example, Riemann gave an explicit formula for the number of primes less than . A prime-yielding formula can be easily constructed from this.
There is also a positive constant , known as Mill's constant, such that is prime for every . It's about if the Riemann hypothesis is correct. It yields the sequence of primes .
Also, the simple recurrence relation generates only primes.
There is no reason why any given kind of "formula for primes" could not exist. It is a popular belief that there is no such thing; I've even heard non-mathematicians claim it, as if it were some kind of established result. I don't believe the OP was asking a silly question at all, because there do exist many such formulae.
Here's another proof that no non-constant polynomial with integer coefficients yields only primes. Let be the constant term of this polynomial. It's quite easy to see that is divisible by for every integer .