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Bunyakovsky conjecture

The Bunyakovsky conjecture is an unsolved problem in number theory, proposed by the Russian mathematician Viktor Bunyakovsky in 1857, which asserts that a polynomial f(x) with integer coefficients takes prime values at infinitely many positive integers n, provided f(x) is irreducible over the rationals, has positive leading coefficient, is primitive (gcd of coefficients is 1), and the values f(n) for positive integers n are collectively coprime (i.e., no fixed prime divides all such values).^[1] This conjecture extends Dirichlet's theorem on arithmetic progressions, which guarantees infinitely many primes for linear polynomials of the form ax + b where \gcd(a, b) = 1, to higher-degree polynomials under the specified conditions.^[1] For example, it predicts that polynomials such as x^2 + 1 or x^2 + x + 41 produce infinitely many primes, though this remains unproven even for quadratics.^[2] The conjecture implies profound results in analytic number theory, such as the infinitude of primes of specific forms, but partial progress has been made only for almost-primes (numbers with bounded prime factors) rather than full primes. Bunyakovsky's statement is a single-polynomial case of the broader Schinzel's hypothesis H from 1960, which addresses simultaneous primality for sets of polynomials without a fixed prime dividing all value tuples. Despite its age and connections to heuristics like the Hardy-Littlewood circle method, the conjecture resists proof due to the difficulty in controlling prime distributions for nonlinear polynomials, with no general resolution expected soon.

Statement and Conditions

Formal Statement

The Bunyakovsky conjecture states that for a polynomial f(x) \in \mathbb{Z} of degree d \geq 1 that is irreducible over the rationals, has positive leading coefficient a_d > 0, and satisfies \gcd(\{f(n) : n \in \mathbb{N}\}) = 1, there are infinitely many natural numbers n such that f(n) is prime.^[3] Irreducibility over the rationals means that f(x) cannot be expressed as a product of two non-constant polynomials with rational coefficients (equivalently, over the integers if primitive). This avoids f(n) factoring non-trivially for all sufficiently large n.^[1] The requirement of a positive leading coefficient a_d > 0 ensures that f(n) > 1 for all sufficiently large natural numbers n, allowing the possibility of primality.^[3] The condition \gcd(\{f(n) : n \in \mathbb{N}\}) = 1 implies that no single prime divides all values of f(n), preventing the sequence from being perpetually composite due to a fixed divisor.^[1] A strong form of the conjecture provides an asymptotic estimate: letting \pi_f(x) denote the number of primes of the form f(n) with f(n) \leq x, then \pi_f(x) \sim c_f \int_2^{x^{1/d}} \frac{dt}{\log f(t)} as x \to \infty, where c_f is the Bateman-Horn constant incorporating the heuristic density from the singular series. This conjecture generalizes Dirichlet's theorem on primes in arithmetic progressions, which establishes the result for the case d=1.^[3]

The Three Conditions

The Bunyakovsky conjecture posits that a polynomial f(x) \in \mathbb{Z} of degree at least 1 produces infinitely many prime values at positive integers n if it satisfies three specific conditions, which are necessary to eliminate structural obstructions to the infinitude of primes. These conditions ensure that the polynomial does not systematically produce composite numbers or values that fail to grow appropriately for primality. The first condition requires that f(x) is irreducible over the rationals. Irreducibility over the rationals means the polynomial cannot be factored into non-constant polynomials of lower degree with rational coefficients. This is essential because if f(x) is reducible, say f(x) = g(x)h(x) where both g and h are non-constant polynomials in \mathbb{Q}, then for sufficiently large positive integers n, both |g(n)| > 1 and |h(n)| > 1, rendering f(n) composite. For instance, the reducible polynomial (x^2 + 1)(x^2 + x + 1) factors into two irreducible quadratics, and its values at large n are products of integers greater than 1 in absolute value, hence composite.^[4] The second condition stipulates that the leading coefficient a_d of f(x) must be positive. With a_d > 0, f(n) grows positively as n increases, ensuring that f(n) > 1 for all sufficiently large positive integers n, which is necessary for these values to be prime candidates. If the leading coefficient were negative, f(n) would eventually become negative for large n, and negative numbers greater than 1 in absolute value are not prime; if zero, the polynomial would not be of the stated degree. This condition thus prevents the polynomial from producing only finitely many positive values exceeding 1. The third condition demands that the greatest common divisor of the values \gcd\{f(n) \mid n \geq 1\} = 1. If this gcd exceeds 1, some prime p divides every f(n), making all such values composite (except possibly finitely many where |f(n)| = p). To compute this gcd, first determine the content of f(x), which is the gcd of its coefficients; then identify any fixed prime divisors p such that f(n) \equiv 0 \pmod{p} for all integers n, which occurs if the reduction of f modulo p vanishes at every residue class modulo p. In practice, for a polynomial of degree d, this gcd equals \gcd(f(0), f(1), \dots, f(d)), as higher values are integer linear combinations of these via finite differences. For example, the polynomial x^2 + x + 2 has content 1 but fixed divisor 2, since it is even for all integer inputs.^[5] These conditions are necessary to remove obvious algebraic or arithmetic barriers that would force all but finitely many f(n) to be composite or non-prime, but they are not sufficient to guarantee infinitude, as the conjecture remains unproven for degrees greater than 1 despite extensive numerical evidence.

Historical Context

Bunyakovsky's Formulation

Viktor Yakovlevich Bunyakovsky (1804–1889) was a prominent Russian mathematician known for his contributions to analysis, probability theory, and number theory.^[6] Born in Bar (now in Ukraine), he studied mathematics in Paris under Augustin-Louis Cauchy and Siméon Denis Poisson, earning his doctorate in 1825 before returning to St. Petersburg, where he became a professor at the university and a member of the Imperial Academy of Sciences.^[6] Bunyakovsky authored over 150 works, including foundational results in integral calculus, such as the integral test for series convergence, and applications in mechanics and hydrostatics.^[6] In 1857, Bunyakovsky formulated his famous conjecture in the paper "Sur les diviseurs numériques invariables des fonctions rationnelles entières," published in the Mémoires de l'Académie Impériale des Sciences de St.-Pétersbourg (6th series, volume VI, pp. 305–329).^[3] This work arose in the context of generalizing earlier results on the distribution of prime numbers, particularly Leonhard Euler's observations on polynomials producing primes and Peter Gustav Lejeune Dirichlet's 1837 theorem proving infinitely many primes in arithmetic progressions under certain conditions.^[7] Bunyakovsky sought to extend these ideas beyond linear polynomials to higher-degree forms, positing a broader principle for when a polynomial with integer coefficients would yield infinitely many prime values at natural number inputs.^[3] Bunyakovsky conjectured that if an irreducible polynomial with integer coefficients and positive leading coefficient satisfies the condition that the greatest common divisor of its values at natural numbers is 1—ensuring it is not constantly divisible by any fixed integer greater than 1—then it represents infinitely many prime numbers.^[7] These three conditions, as originally outlined by Bunyakovsky, were proposed as both necessary and sufficient for the infinitude of primes generated by the polynomial, drawing an analogy to the arithmetic progressions in Dirichlet's theorem.^[3] His formulation highlighted the potential for systematic prime production via polynomials.

Early Developments and Influences

The observation of Leonhard Euler in the 18th century that the quadratic polynomial x^2 + x + 41 yields prime numbers for 40 consecutive integer values x = 0, 1, \dots, 39 highlighted the potential of polynomials to generate sequences of primes, laying early groundwork for later conjectures on prime-producing polynomials.^[8] Bunyakovsky's 1857 formulation directly extended Dirichlet's 1837 theorem, which established the infinitude of primes in arithmetic progressions and corresponds to the linear case of the conjecture where the polynomial is of degree one.^[9] This influence underscored the conjecture's roots in analytic number theory, shifting focus from linear forms to higher-degree irreducible polynomials satisfying analogous irreducibility and coprimality conditions.^[4] By the 1920s, the problem was widely recognized as unsolved for degrees greater than one, with initial attempts highlighting the challenges beyond the linear regime. Carl Ludwig Siegel's investigations in the 1930s advanced bounds on exceptional zeros of Dirichlet L-functions, providing effective estimates that refined the error terms in prime distribution results for arithmetic progressions.^[10] Concurrently, G. H. Hardy and J. E. Littlewood applied the circle method to develop heuristic asymptotics for the count of prime values taken by polynomials, predicting densities consistent with Bunyakovsky's claim and influencing subsequent probabilistic approaches to the problem.^[11]

Examples

Quadratic Polynomials

One prominent example of a quadratic polynomial satisfying the conditions of the Bunyakovsky conjecture is Euler's polynomial f(x) = x^2 + x + 41. This polynomial is irreducible over the integers, as its discriminant $1 - 164 = -163 is negative and square-free, ensuring no rational roots by the rational root theorem. It has a positive leading coefficient of 1, and the greatest common divisor of its values f(n) for integer n is 1, since no prime divides all such values—for instance, f(0) = 41 is prime. Remarkably, f(n) yields prime numbers for 40 consecutive nonnegative integers n = 0 to $39, such as f(0) = 41, f(1) = 43, and f(39) = 1601 (which is prime), but f(40) = 1681 = 41^2 is composite. A related quadratic achieving a longer streak is g(x) = x^2 - 79x + 1601, which produces primes for 80 consecutive values x = 0 to $79. This polynomial also meets the conjecture's conditions: it is irreducible (discriminant $79^2 - 4 \cdot 1601 = 6241 - 6404 = -163, the same as Euler's), has leading coefficient 1, and gcd of values is 1. In fact, g(x) is a shifted version of Euler's polynomial, obtained by substituting x \to x - 40, highlighting how transformations can extend prime streaks while preserving the underlying properties. Another classic quadratic example is h(x) = x^2 + 1, which is irreducible over the integers (no real roots beyond imaginaries), has leading coefficient 1, and gcd of values is 1, as the sequence includes primes like h(1) = 2, h(4) = 17, and h(6) = 37, with no fixed prime divisor. Although it is unknown whether h(n) produces infinitely many primes, numerical computations show it generates primes sporadically for larger n, such as h(10) = 101 and h(14) = 197. These examples illustrate the phenomenon of prime streaks in quadratic polynomials under the conjecture's conditions, where the longest known consecutive run remains 80 for g(x). Heuristically, the density of primes among f(n) for a quadratic f is expected to be approximately \frac{1}{\log f(n)} \sim \frac{2}{\log n}, reflecting the prime number theorem's influence on polynomial values.

Higher-Degree and Cyclotomic Polynomials

The polynomial f(x) = x^3 + x^2 - 2x - 1 provides a notable cubic example satisfying the conditions of the Bunyakovsky conjecture. It is irreducible over the rationals, as it serves as the minimal polynomial for $2\cos(2\pi/7)^[12]. The leading coefficient is positive, and the greatest common divisor of its values at all integers is 1, since f(0) = -1. For small positive integers x, it yields prime values, including f(2) = 7, f(3) = 29, f(4) = 71, f(5) = 139, and f(6) = 239. Cyclotomic polynomials \Phi_n(x) for n > 1 also meet the irreducibility and positive leading coefficient requirements, being monic and irreducible over the rationals by Gauss's theorem on cyclotomic polynomials. The gcd condition holds, as \Phi_n(0) = 1 for n > 1, ensuring no fixed prime divisor. For odd primes p, \Phi_p(x) = \frac{x^p - 1}{x - 1} = x^{p-1} + x^{p-2} + \cdots + x + 1 represents a specific case, with \Phi_p(1) = p. A concrete illustration is \Phi_5(x) = x^4 + x^3 + x^2 + x + 1, which produces primes such as \Phi_5(1) = 5 and \Phi_5(2) = 31^[13]. In higher degrees, polynomials satisfying the conjecture's conditions tend to grow more rapidly than their lower-degree counterparts, resulting in larger values even for modest inputs and thus fewer observed small primes. Despite this, the Bunyakovsky conjecture posits that infinitely many primes arise from such polynomials for integer arguments.

Partial Results and Evidence

Dirichlet's Theorem for Linear Case

Dirichlet's theorem on arithmetic progressions, proved by Peter Gustav Lejeune Dirichlet in 1837, asserts that if a and d are positive coprime integers, then there are infinitely many prime numbers congruent to a modulo d.^[14] This result corresponds precisely to the case of linear polynomials in the Bunyakovsky conjecture, where f(x) = dx + a is a degree-1 polynomial that is automatically irreducible over the integers, has positive leading coefficient d > 0, and satisfies the fixed divisor condition \gcd_{n \geq 1} f(n) = 1 if and only if \gcd(a, d) = 1.^[4] The proof of Dirichlet's theorem introduces Dirichlet characters modulo d—multiplicative functions \chi: \mathbb{Z} \to \mathbb{C} periodic with period d—and the associated L-functions L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s}. By analyzing the Euler product for L(s, \chi) and showing that L(1, \chi) \neq 0 for all non-principal characters \chi, Dirichlet established that the sum \sum_{p \equiv a \pmod{d}} 1/p diverges, implying the infinitude of such primes.^[15] This theorem directly verifies the Bunyakovsky conjecture for linear polynomials, including the special case d=1 where f(x) = x + a produces infinitely many primes. The prime number theorem for arithmetic progressions further quantifies the distribution, stating that the primes congruent to a modulo d have asymptotic density $1/\phi(d) among all primes, where \phi is Euler's totient function.^[16] Proved two decades before Bunyakovsky's 1857 generalization to higher degrees, Dirichlet's theorem provided essential motivation for exploring prime values of polynomials.^[17]

Numerical Evidence and Bounds for Higher Degrees

Computational studies provide substantial numerical evidence supporting the Bunyakovsky conjecture for polynomials of degree greater than 1. For the quadratic polynomial f(x) = x^2 + x + 41, which satisfies the conjecture's conditions, evaluations up to x = 10^6 yield approximately $10^5 prime values, demonstrating a persistent production of primes consistent with heuristic expectations. Similar results hold for other quadratics, such as f(t) = 32t^2 + 20t + 1, where computations up to t = 10^8 identify over 12 million prime values, closely matching the predicted count from asymptotic formulas with a relative error below 0.05%.^[18] For higher-degree polynomials, sieving techniques have verified the generation of numerous primes up to substantial bounds. For instance, the cubic polynomial f(x) = x^3 - x^2 + 1 produces thousands of prime values for x up to $10^6, with no fixed prime divisor obstructing further occurrences. These computations, enabled by advanced algorithms for primality testing and sieving, align with the conjecture's predictions and show no signs of exhaustion in prime production.^[19] Theoretical bounds offer further support under additional assumptions. Assuming the Generalized Riemann Hypothesis (GRH), effective versions of the conjecture guarantee the existence of primes among f(n) for n up to x = \exp(c (\log f(1))^k), where c and k depend on the degree and coefficients, providing explicit limits beyond which primes must appear. Unconditionally, analogs of Linnik's theorem yield weaker bounds, ensuring primes in polynomial sequences within polynomial ranges of the height, though these are less sharp for degrees greater than 1. Heuristic models, notably the Bateman-Horn conjecture, predict the asymptotic density of primes generated by such polynomials as approximately c \int_2^x \frac{dt}{(\log t)^d}, where d is the degree and c > 0 is the singular series reflecting local densities modulo primes. This formula, a refinement of earlier Hardy-Littlewood heuristics, has been validated by numerical data across various degrees, with no counterexamples observed. As of 2025, extensive searches confirm that every irreducible polynomial satisfying the Bunyakovsky conditions continues to produce primes beyond small n, bolstering confidence in the conjecture's validity.

Schinzel's Hypothesis H

Schinzel's Hypothesis H, proposed by Andrzej Schinzel in 1960 in a joint paper with W. Sierpiński and later elaborated in his work on polynomials, is a broad generalization of the Bunyakovsky conjecture to systems of multiple polynomials.^[20] Specifically, let f_1(x), \dots, f_k(x) \in \mathbb{Z} for k \geq 1 be irreducible polynomials over the rationals with integer coefficients and positive leading coefficients. The hypothesis asserts that if there is no fixed prime p dividing the product f_1(n) \cdots f_k(n) for all natural numbers n (equivalently, for every prime p, there exists an integer n such that p divides none of the values f_i(n)), then there are infinitely many natural numbers n such that all f_i(n) are prime numbers. This formulation directly implies Bunyakovsky's conjecture as the special case k=1, reducing to the conditions of irreducibility, positive leading coefficient, and no fixed prime divisor for the single polynomial. The generalized Bunyakovsky conjecture and the generalized Dickson's conjecture are equivalent to Hypothesis H, extending the original to multiple polynomials under the same local-global condition that prevents sieve obstructions.^[21] For linear polynomials (\deg f_i = 1), it specializes to Dickson's conjecture, predicting infinitely many n such that all linear forms a_i n + b_i (with \gcd(a_i, b_i)=1 and the no fixed divisor condition) yield primes simultaneously. The hypothesis encompasses longstanding problems like the twin prime conjecture (f_1(x) = x, f_2(x) = x + 2) and Sophie Germain primes (f_1(x) = x, f_2(x) = 2x + 1), both cases where k=2 and linear polynomials satisfy the conditions with no fixed prime dividing n(n+2) or n(2n+1) for all n. For k > 1 and higher degrees, the challenge intensifies due to potential correlations between the polynomials leading to shared prime factors. Extensions of Hypothesis H have implications in analytic number theory. Assuming H, one derives strong forms of the prime tuples conjecture, including bounded gaps in admissible prime constellations, extending results like those of Zhang and Maynard to denser sets. Connections to the abc conjecture provide height estimates yielding positive densities for prime values in polynomial families, while Vojta's conjectures offer Diophantine approaches to simultaneous primality. As of November 2025, Schinzel's Hypothesis H remains open in full generality over the integers, with no complete proofs even for small k and low degrees beyond Dirichlet's theorem. Partial progress includes unconditional resolutions in multivariable settings over polynomial rings and function fields (where irreducibility replaces primality), conditional results under the Generalized Riemann Hypothesis for small k (e.g., almost-primes for k=2 linear polynomials), and the establishment of the hypothesis for 100% of polynomials on average (Helfgott and Serre, 2022). Recent advances in 2025, such as improvements on the coprime variant of the hypothesis (Buium et al., arXiv:2502.09959), further support its role in local-global principles for conic bundles, though a general resolution over \mathbb{Z} is not expected soon.^[22]^[23]

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