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# the Bateman–Horn Conjecture?

Communicated by *Notices* Associate Editor William McCallum

For a given family of univariate polynomials with integer coefficients, how often should we expect their values at positive integer arguments to be simultaneously prime? The Bateman–Horn conjecture, first formulated by Paul T. Bateman and Roger A. Horn in 1962

To understand where the Bateman–Horn conjecture comes from, we start with the prime number theorem. The exposition below follows

### Prime number theorem

Let denote the number of primes at most The prime number theorem, proved independently by Hadamard and de la Vallée Poussin in 1896, says that . in which ,

1is the *logarithmic integral* and is asymptotic equivalence; that is, means .

The prime number theorem suggests the possibility of a random model for the prime numbers: the probability that is prime is about The Bateman–Horn conjecture follows by pursuing this to its logical extreme, while adjusting for congruence obstructions (for example, . is the only even prime).

### A single polynomial

Let denote the set of polynomials in with coefficients in the set of integers. For , define ,

in which denotes the cardinality of a set and is a natural number. What conditions must satisfy if it generates infinitely many distinct primes?

First, should be nonconstant and its leading coefficient must be positive. Second, should be irreducible in Less obvious is that . should not vanish identically modulo any prime. For example, is irreducible, but so , is always divisible by .

Suppose is nonconstant, irreducible, and does not vanish identically modulo any prime. Let and suppose that has leading coefficient Then . and our heuristic suggests that the probability is prime is

2so we expect that

3However, this is incorrect since we failed to take into account how likely it is that (the letter will always denote a prime number). If we assume for the sake of our heuristic argument that divisibility by distinct primes are independent events, then we should weight our prediction by

4in which is the number of solutions to since , is the probability that is divisible by and is the probability that a random integer is divisible by Thus, for a single polynomial . we suspect that ,

### Multiple polynomials

Suppose are distinct, nonconstant, irreducible polynomials with positive leading coefficients. Although maybe no single vanishes identically modulo a prime, the product might. For example, neither nor vanish identically modulo any prime, but their product vanishes identically modulo This “congruence obstruction” prevents . and from being simultaneously prime infinitely often. Consequently, we must require that does not vanish identically modulo any prime.

Reasoning as above suggests the probability that all of the are simultaneously prime is

Thus, the expected number of such that are prime is around

As before, this prediction is off by a constant factor. Instead of dividing by in 4, we now divide by the probability that a randomly selected , of integers has no element divisible by -tuple .

### The conjecture

Putting this all together yields the final conjecture (the convergence of the infinite product below is not obvious; see

Only a few special cases of the conjecture, such as the prime number theorem for arithmetic progressions, are known to be true. However, an upper bound comparable to the conjectured asymptotic is provided by the Brun sieve

### Applications

Landau asked if there are infinitely many primes of the form The Bateman–Horn conjecture with . suggests that the answer is yes. Indeed,

where so , and we expect that see Figure ;1.

Applying the Bateman–Horn conjecture to and suggests the truth of the twin-prime conjecture. Indeed, and are simultaneously prime if and only if is the least prime in a twin-prime pair. For ,

so the corresponding Bateman–Horn constant is

in which is the *twin primes constant*. The Bateman–Horn conjecture predicts that

In fact, we get the same prediction for and with .

A *Sophie Germain prime* is a prime such that is prime. The Bateman–Horn conjecture with and yields the same prediction as in the twin-prime case; see Figure 2. The Bateman–Horn even explains the presence of curious patterns in the Ulam spiral

## References

[ AZFG20] - Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia,
*The Bateman-Horn conjecture: heuristic, history, and applications*, Expo. Math.**38**(2020), no. 4, 430–479, DOI 10.1016/j.exmath.2019.04.005. MR4177951,## Show rawAMSref

`\bib{Expo}{article}{ author={Aletheia-Zomlefer, Soren Laing}, author={Fukshansky, Lenny}, author={Garcia, Stephan Ramon}, title={The Bateman-Horn conjecture: heuristic, history, and applications}, journal={Expo. Math.}, volume={38}, date={2020}, number={4}, pages={430--479}, issn={0723-0869}, review={\MR {4177951}}, doi={10.1016/j.exmath.2019.04.005}, }`

[ BH62] - Paul T. Bateman and Roger A. Horn,
*A heuristic asymptotic formula concerning the distribution of prime numbers*, Math. Comp.**16**(1962), 363–367, DOI 10.2307/2004056. MR148632,## Show rawAMSref

`\bib{Bateman}{article}{ author={Bateman, Paul T.}, author={Horn, Roger A.}, title={A heuristic asymptotic formula concerning the distribution of prime numbers}, journal={Math. Comp.}, volume={16}, date={1962}, pages={363--367}, issn={0025-5718}, review={\MR {148632}}, doi={10.2307/2004056}, }`

[ BH65] - Paul T. Bateman and Roger A. Horn,
*Primes represented by irreducible polynomials in one variable*, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, RI, 1965, pp. 119–132. MR176966,## Show rawAMSref

`\bib{BatemanHorn-2}{article}{ author={Bateman, Paul T.}, author={Horn, Roger A.}, title={Primes represented by irreducible polynomials in one variable}, conference={ title={Proc. Sympos. Pure Math., Vol. VIII}, }, book={ publisher={Amer. Math. Soc., Providence, RI}, }, date={1965}, pages={119--132}, review={\MR {176966}}, }`

[ HL23] - G. H. Hardy and J. E. Littlewood,
*Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes*, Acta Math.**44**(1923), no. 1, 1–70, DOI 10.1007/BF02403921. MR1555183,## Show rawAMSref

`\bib{Hardy}{article}{ author={Hardy, G. H.}, author={Littlewood, J. E.}, title={Some problems of `Partitio numerorum'; III: On the expression of a number as a sum of primes}, journal={Acta Math.}, volume={44}, date={1923}, number={1}, pages={1--70}, issn={0001-5962}, review={\MR {1555183}}, doi={10.1007/BF02403921}, }`

[ SS58] - A. Schinzel and W. Sierpiński,
*Sur certaines hypothèses concernant les nombres premiers*(French), Acta Arith.**4**(1958), 185–208; erratum 5 (1958), 259, DOI 10.4064/aa-4-3-185-208. MR106202,## Show rawAMSref

`\bib{Schinzel}{article}{ author={Schinzel, A.}, author={Sierpi\'{n}ski, W.}, title={Sur certaines hypoth\`eses concernant les nombres premiers}, language={French}, journal={Acta Arith.}, volume={4}, date={1958}, pages={185--208; erratum 5 (1958), 259}, issn={0065-1036}, review={\MR {106202}}, doi={10.4064/aa-4-3-185-208}, }`

[ Ten15] - Gérald Tenenbaum,
*Introduction to analytic and probabilistic number theory*, 3rd ed., Graduate Studies in Mathematics, vol. 163, American Mathematical Society, Providence, RI, 2015. Translated from the 2008 French edition by Patrick D. F. Ion, DOI 10.1090/gsm/163. MR3363366,## Show rawAMSref

`\bib{Tenenbaum}{book}{ author={Tenenbaum, G\'{e}rald}, title={Introduction to analytic and probabilistic number theory}, series={Graduate Studies in Mathematics}, volume={163}, edition={3}, note={Translated from the 2008 French edition by Patrick D. F. Ion}, publisher={American Mathematical Society, Providence, RI}, date={2015}, pages={xxiv+629}, isbn={978-0-8218-9854-3}, review={\MR {3363366}}, doi={10.1090/gsm/163}, }`

## Credits

Figures are courtesy of Stephan Ramon Garcia.

Author photo is courtesy of Gizem Karaali.