Quadratic expression that generate primes

Question

I recently learned that there exist quadratic expression that generate some primes and some of these equations generate more primes than others. In the following video, the person shows the following expression

$$4x^2 -2x+1$$

that generates some primes. So I was wondering, are there infinitely many expression that exist that can produce primes and what makes one quadratic produce more primes than another? In other words, do some quadratic expressions have a special property that causes them to generate more primes than others?

Thanks!

Answer 1

There are infinitely many linear polynomials $f(n)=an+b$ such that $f(n)$ yields prime values at infinitely many integer arguments - I assume this is what "producing primes" means. There are also infinitely many quadratic polynomials that obtain at least one prime value, e.g. $x^2+ax+p$ for various integers $a$ and primes $p$ at $x=0$. Whether or not we know of any quadratics that provably obtain an infinite number of prime values at integer arguments, I'm not sure.

If increase the degree and number of variables, though, we do know of such polynomials that take infinitely many prime values, in fact whose positive values are only prime numbers! See this section of Wikipedia for more information.

Beyond linear polynomials (the subject of the quantitative version of Dirichlet's theorems on primes in arithmetic progressions, which generalizes in a very different direction in algebraic number theory on Cheboratev density), capturing the asymptotic frequency with which systems of polynomials simultaneously yield primes is a very hard and at the moment speculative business.

Bateman-Horn conjecture. Let $f_i(x)$ be a finite family of irreducible polynomials (say with positive leading coefficients). Let $P(n)$ count $k\le n$ for which $f_i(k)$ are prime for all $i$. Then $$P(n)\sim \frac{\displaystyle \prod_p \frac{1-N(p)/p}{(1-1/p)^m}}{(\deg f_1)\cdots(\deg f_m)}\int_2^n\frac{dt}{(\log t)^m}$$ where $N(p)$ counts the solutions to $\{ f_i(x)=0$ in ${\bf F}_p$ (the finite field of integers mod $p$).

• A heuristic asymptotic formula concerning the distribution of prime numbers

"Heuristic" here - ubiquitous in analytic number theory - effectively means techniques based on experience, empirical data, intuition, educated guessing, and informal statistical reasoning.

Note that a very particular case of Bateman-Horn is the quantitative version of the twin prime conjecture, the $1$st Littlewood-Hardy conjecture. It also implies the Bunyakovsky_conjecture, the one-polynomial case of BH, and is a quantitative refinement of Shinzel's hypothesis, itself an extension of Bunyakovsky to multiple polynomials. Thus BH is a massively general statement.

Even so, I do not think any particular case of BH has been proven. That includes any quadratic polynomials, which are the specific subject of your question. As for intuition as to why some expressions may generate them more frequently then others (detectable in the difference in the multiplicative constants in the giant formula above), the form of the conjecture suggests that the answer may lie in local-global thinking (another thing that turns up a lot in number theory).

That is to say that there may be good reason to expect that long-term "global" nature of $P(n)$ can be reliably statistically forecasted by the "local" nature of it (the counting function $N(p)$). Given this expectation, "polynomials yield primes at different rates" reduces to "polynomials don't always have the same number of zeros mod every prime $p$," which seems (to me at least) to be not quite as surprising (since it is more finitistic, I guess).