Editing Bateman–Horn conjecture (section)

==Examples==
If the system of polynomials consists of the single polynomial ''ƒ''<sub>1</sub>(''x'')&nbsp;=&nbsp;''x'', then the values ''n'' for which ''ƒ''<sub>1</sub>(''n'') is prime are themselves the prime numbers, and the conjecture becomes a restatement of the [[prime number theorem]].

If the system of polynomials consists of the two polynomials ''ƒ''<sub>1</sub>(''x'')&nbsp;=&nbsp;''x'' and ''ƒ''<sub>2</sub>(''x'')&nbsp;=&nbsp;''x''&nbsp;+&nbsp;2, then the values of ''n'' for which both ''ƒ''<sub>1</sub>(''n'') and ''ƒ''<sub>2</sub>(''n'') are prime are just the smaller of the two primes in every pair of [[twin prime]]s. In this case, the Bateman–Horn conjecture reduces to the [[Twin prime#First Hardy–Littlewood conjecture|Hardy–Littlewood conjecture]] on the density of twin primes, according to which the number of twin prime pairs less than ''x'' is
:<math>\pi_2(x) \sim 2  \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2}\frac{x}{(\log x)^2 } \approx 1.32 \frac {x}{(\log x)^2}.</math>