New way of finding primes - Can it be proven wrong?

Question

So, I believe I may have discovered a new way for finding primes, but I'm not sure if there is a definite proof; If not, then I have just created my own one.

Anyway, this new method I've come up with is that you take any integer above $0$, $x$, and times it by $1$ and add $1$, then times it by $2$ and add $1$, then times it by $3$ and add $1$, etc, all the way up to $(x^{2} +1)$

$(x +1), (2x +1), (3x +1), (4x +1), ... ,(x^{2} +1)$

My conjecture is that at least one of these terms will give you a prime number, when $x$ is any integer above $0$.

Is there any way to prove me right/wrong?

Answer 1

This site

https://en.wikipedia.org/wiki/Linnik%27s_theorem

shows what has been found out for the least prime in an arithmetic progression. The conjecture $$p(a,d)<d^2$$ would imply that you can always find a prime this way, but this has not been unconditionally proven.

For Goldbach's conjecture we need two numbers which are simultaneously prime, so even if we can always find primes this way, there is no guarantee that they can be summed up to a given even number. I cannot see a way to use this approach for solving Goldbach's conjecture.

Answer 2

$$n\equiv-(x^{-1})\bmod p$$ are knocked out as lead coefficients. They create a number divisible by p any time p is not a factor of x. That means just 57 survive for x=100, p<10 and only 23 survive that for x=101 . Accounting for overlap is really the only tricky part for me. I count it wrong without code.