Cilleruelo's conjecture on the least common multiple of polynomial sequences

For a polynomial $f\in\mathbb{Z}[x]$ irreducible over the rationals with degree $d\ge2$, write $L_f(N)$ for the least common multiple of the values $|f(1)|,\dots,|f(N)|$. Cilleruelo's conjecture (2011) asserts $\log L_f(N)\sim(d-1)N\log N$ as $N\to\infty$. The quadratic case is Cilleruelo's theorem; for every degree $d\ge3$ the conjecture is open, and no single irreducible polynomial of degree above two is known to satisfy it. A matching upper bound $\log L_f(N)\le(d-1+o(1))N\log N$ is known, so the problem is equivalent to a sharp lower bound. The Maynard-Rudnick lower-bound paper (2019/2021) proved the first bound of the conjectured order with constant of order $1/d$; Sah (2020) raised the constant to 1, Technau (2025) obtained $2(d-1)/d$ for monic even polynomials, and Entin (2025) obtained $(d-1)/(d-r)$ for polynomials of degree $d\ge4$ whose roots partition into $r$ potent subsets (a class that includes the decomposable polynomials). All known constants remain a factor growing with $d$ below the conjectured one. The surviving open core is the asymptotic, equivalently the constant-$(d-1)$ lower bound, for any and all degrees $d\ge3$.

Background

The least common multiple (lcm) of a finite set of positive integers is the smallest positive integer divisible by all of them; it records which primes, and to which powers, occur in the collection. For a polynomial $f\in\mathbb{Z}[x]$ set

$$L_f(N)=\operatorname{lcm}\big(|f(1)|,|f(2)|,\dots,|f(N)|\big),$$

the lcm of the first $N$ values of $f$ (absolute values make this well-defined for negative values; an irreducible polynomial of degree at least two has no integer root, so no value vanishes). The model case is the identity $\log\operatorname{lcm}(1,\dots,N)=\psi(N)\sim N$, where $\psi$ is Chebyshev's function (the sum of $\log p$ over all prime powers $p^k\le N$) and the asymptotic is equivalent to the prime number theorem. For polynomials built from linear factors the growth of $\log L_f(N)$ remains linear, with constants expressible through primes in arithmetic progressions: Bateman, Kalb and Stenger treat arithmetic progressions, and Hong, Qian and Tan treat sequences of products of linear polynomials. The situation changes for irreducible $f$ of degree $d\ge2$: the values $f(n)$ with $1\le n\le N$ have size about $N^d$, so their product satisfies $\log\prod_{n\le N}|f(n)|=dN\log N+O(N)$, while the lcm discards prime factors shared by different values; determining $L_f(N)$ therefore measures the multiplicative independence of polynomial values. Cilleruelo settled the quadratic case completely, proving $\log L_f(N)=N\log N+B_fN+o(N)$ with an explicit constant $B_f$ depending on $f$, and conjectured the general law now known as Cilleruelo's conjecture:

$$\log L_f(N)\sim(d-1)N\log N\qquad(N\to\infty)$$

for every $f\in\mathbb{Z}[x]$ irreducible over $\mathbb{Q}$ of degree $d\ge2$. The heuristic behind the constant $d-1$ is that primes $p\le N$ contribute about $N\log N$ in total to $\log L_f(N)$: the number of $n\le N$ with $p\mid f(n)$ is close to $\rho_f(p)N/p$, where $\rho_f(p)$ is the number of roots of $f$ modulo $p$, and the Chebotarev density theorem (a refinement of Dirichlet's theorem on primes in progressions describing how primes split in number fields) gives $\sum_{p\le N}\rho_f(p)\log p/(p-1)=\log N+O(1)$. The conjecture asserts that beyond this small-prime part essentially the entire remaining $(d-1)N\log N$ of the product survives in the lcm, i.e. that the parts of the values supported on primes above $N$ are nearly coprime across different $n$. Two facts delimit what is known. First, the upper bound matching the conjecture is known: an examination of Cilleruelo's argument shows $\log L_f(N)\le(d-1+o(1))N\log N$ for every irreducible $f$ of degree $d\ge3$, so the conjecture is equivalent to the lower bound $\log L_f(N)\ge(d-1-o(1))N\log N$. Second, lower bounds of the correct order $N\log N$ are recent. Before 2019 only bounds of order $N$ were known (Hong, Luo, Qian and Wang proved $\log L_f(N)\gg N$ for polynomials with non-negative integer coefficients, with subsequent refinements by Hong and Qian). Maynard and Rudnick then proved the first bound of the conjectured order for every irreducible $f$ of degree $d\ge2$, with constant of order $1/d$: writing $P^+(m)$ for the largest prime factor of $m$, they used that $P^+(f(n))>n$ for a positive proportion of $n$ (in fact for at least a proportion $1-1/d$), together with the fact that a prime $p>N/\log N$ can be the largest prime factor $P^+(f(m))$ for at most $d$ values $m\le N$ (because $f$ has at most $d$ roots modulo $p$); hence the values $f(n)$ contain $\gg N/d$ distinct primes exceeding $N/\log N$, each contributing at least $\log(N/\log N)$ to $\log L_f(N)$. Sah reorganized the argument around the total product $Q_f(N)=\prod_{n\le N}|f(n)|$: after removing the small-prime part and using a polynomial identity to bound how often a power of a large prime can divide values, each surviving large prime is counted with multiplicity, which yields the constant 1 (independent of $d$) for the lcm, the constant $2/d$ for the radical $\ell_f(N)=\operatorname{rad}(L_f(N))$ (the product of the distinct primes dividing $L_f(N)$), and the companion conjecture $\log\ell_f(N)\sim(d-1)N\log N$. All subsequent improvements concern special classes of polynomials. Technau treats even polynomials (those with $f(x)=f(-x)$): for monic irreducible even $f$ of degree $d\ge2$ he proves $\log L_f(N)\ge\frac{2(d-1)}{d}N\log N-O(N)$, using zero-sum relations amongst the roots of $f$ that restrict how a large prime can divide several values. Entin proves $\log L_f(N)\ge(\frac{d-1}{d-r}-o(1))N\log N$ (for irreducible $f$ of degree $d\ge4$) whenever the set of complex roots of $f$ can be partitioned into $r$ potent subsets, where a finite set of algebraic numbers is called potent if some positive-integer weighted sum of its elements is an integer; this applies in particular to irreducible decomposable polynomials $f=g\circ h$ (compositions with $g,h\in\mathbb{Z}[x]$ of degree at least two), whose root fibers over the roots of $g$ are potent by Vieta's formulas, and the even case corresponds to $h(x)=x^2$. Entin also improves the general radical constant to $(d-1)/V_d$ with $V_d=\sum_{1\le k\le\min(\lfloor 3d/4\rfloor,d-2)}(d-k)$, a constant of size about $32/(15d)$ for large $d$. A function-field analogue over $\mathbb{F}_q[T]$ (polynomials over a finite field, with monic polynomials of fixed degree in place of the integers $1,\dots,N$) has been settled by Entin and Landsberg for a class of special polynomials that includes all quadratics and many of higher degree; there the lcm is also asymptotic to its radical. The methods rely on exact degree-based counting in characteristic $p$ and do not transfer to the integers. For degree $d=3$ the best known constant is 1 against the conjectured 2; in general the gap is the factor $d-1$; for even polynomials it is the factor $d/2$; and no explicit polynomial of degree above two, neither a cubic such as $x^3+2$ nor the cyclotomic family $X^{2^\eta}+1$, is known to satisfy the conjecture.

Problem Statement

Prove or disprove Cilleruelo's conjecture: for every polynomial $f\in\mathbb{Z}[x]$ that is irreducible over $\mathbb{Q}$ and has degree $d\ge2$,

$$\log L_f(N)\sim(d-1)N\log N\qquad(N\to\infty),$$

where $L_f(N)=\operatorname{lcm}(|f(1)|,\dots,|f(N)|)$. The case $d=2$ is Cilleruelo's theorem, so the open content is $d\ge3$. Since the matching upper bound $\log L_f(N)\le(d-1+o(1))N\log N$ holds for every irreducible $f$ of degree $d\ge3$, the problem is equivalent to proving the lower bound $\log L_f(N)\ge(d-1-o(1))N\log N$ for every such $f$. A valid resolution is either a proof of the asymptotic for all irreducible polynomials of degree $d\ge3$, or an explicit irreducible polynomial of some degree $d\ge3$ together with a rigorous proof that the asymptotic fails for it. Establishing the asymptotic for even one polynomial of degree $d\ge3$ would be the first known example and major progress, but does not by itself resolve the conjecture as stated.

References

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14. Ilaria Viglino, 'An average version of Cilleruelo's conjecture for families of $S_n$-polynomials over a number field', preprint arXiv:2402.03999, https://arxiv.org/abs/2402.03999