Problem 976

../

claims/: The 9 claim pages of Problem 976, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\leq m\leq n$ with $f(m)$ is divisible by a prime $\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of

$$\prod_{1\leq m\leq n}f(m).$$

Estimate $F_f(n)$. In particular, is it true that $F_f(n)\gg n^{1+c}$ for some constant $c>0$? Or even $\gg n^d$?

Formulation. The questions are read for each fixed $f$: does $F_f(n)\gg_f n^{1+c_f}$ hold for some $c_f>0$, or even $F_f(n)\gg_f n^d$? This records the asymptotic questions on the live catalog page, last edited 1 February 2026. The site says "some constant $c>0$" after fixing $f$; it does not explicitly require one exponent to work for all polynomials. The stronger quantifier order $\exists c>0\,\forall f$ is separate. Constants and thresholds here may depend on the fixed polynomial. Read greatest prime factors on absolute values, with $P^+(1)=1$ for any small unit products. Irreducibility and $d\geq2$ exclude zero factors, and the products have absolute value greater than one eventually.

Status. Open. The site labels the problem OPEN (page last edited 1 February 2026). Neither the general fixed-$f$ power-gain question nor the stronger degree-scale question is resolved for every irreducible polynomial: the general subpower theorem and the special-family power bounds below fall short of both. Bhalla's conditional note of 2026-04-16 gives the degree-scale bound only under an unproved prime-values hypothesis. The special-family claims answer the first question for particular polynomials: accepted for $t^3+2$ (Heath-Brown, Irving), for even Klein-group quartics (de la Bretèche), for cyclic and dihedral quartics (Dartyge--Maynard) and for $t^2+1$ (Pascadi); pending for monic cubics (Ermoshin), for $at^2+h$ (Grimmelt--Merikoski) and for $t^2+1$ (Carella). None covers every irreducible polynomial, so both questions stay open.

Source. erdosproblems.com/976, accessed 2026-09-09. Cite as: T. F. Bloom, Erdős Problem #976, https://www.erdosproblems.com/976.

References.

• [Er52c] Erdős, P., On the greatest prime factor of $\prod^x_{k=1}f(k)$. J. London Math. Soc. (1952), 379-384.
• [Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
• [ErSc90] Erdős, P. and Schinzel, A., On the greatest prime factor of $\prod^x_{k=1}f(k)$. Acta Arith. (1990), 191-200.
• [Na22] Nagell, T., Zur Arithmetik der Polynome. Abhandlungen aus dem Math. Seminar Hamburg (1922), 179-194.
• [Ri34] Ricci, G., Su un teorema di Tchebychef-Nagel. Annali di Mat. (1934), 295-303.
• [Te90] Tenenbaum, Gérald, Sur une question d'Erdős et Schinzel. (1990), 405-443.

These six catalog references are retained as historical bibliography.

Formalization. None recorded.

Current assessment

On 2026-09-09 the site labeled the problem OPEN with no proof claim. The sources below are the results found as of that date: the general subpower bound, the special-family papers, and Bhalla's forum announcement of a conditional note. None of them gives an unconditional general resolution.

The Dartyge--Maynard and Grimmelt--Merikoski statements are cited from their preprints. No theorem detail is adopted from the older Erdős--Schinzel digest.

On 2026-10-05 the site labeled the problem OPEN (last edited 1 February 2026); its thread held three comments, the newest of 16 April 2026 (Bhalla's announcement and a reply reporting a check that found no issue), and no proof claim; the community database listed the problem open; neither conjectures.io nor Palomar had an entry. The community's AI-contributions wiki (frozen with data to 30 June 2026) lists Bhalla's note of 16 April 2026 as a conditional partial result, so the route of the conditional-route lead below has been public since that date. Nothing found gives $F_f(n)\gg_f n^{1+c}$ for every irreducible $f$ or the degree-scale bound. Carella's unrefereed arXiv:2308.10075 (first posted 19 August 2023, v7 of 13 June 2025) claims $x^{3/2}$ for the dyadic product of $n^2+1$, an exponent above every published one; it is recorded as a pending claim. A formal-conjectures pull request to formalize the statement (5873) was closed unmerged on 2026-09-13.

Known Results

General progress

Erdős's 1952 Theorem, display (2) gives, for each fixed irreducible $f$ of degree greater than one and all sufficiently large $n$,

$$F_f(n)>n(\log n)^{c_2(f)\log\log\log n}, \qquad c_2(f)>0.$$

Printed p. 379 explicitly reduces to this safe irreducible class. Its broader opening wording permits zero products, for example from $(x-1)(x^2+1)$, so that wording is not used as a general theorem domain here. The same page credits Nagell with the preceding $n\log n$ bound. Display (3) asserts the stronger $n\exp\{(\log n)^{c_3(f)}\}$ shape, but printed p. 380 explicitly withholds its proof. It receives no proved-result credit from that paper.

The strongest general bound found is Tenenbaum II, Theorem 2 (Inventiones Mathematicae 99 (1990), printed p. 216):

$$F_f(n)>n\exp\{(\log n)^\alpha\}, \qquad 0<\alpha<2-\log4.$$

Here $f$ is irreducible of degree greater than one; $\alpha$ is fixed before $n$ tends to infinity, without uniformity in $\alpha$ asserted. The June 2026 introduction of Ermoshin's arXiv v3 also identifies this general bound. Since every permitted $\alpha$ is less than one, the extra factor is $n^{o(1)}$ and does not supply any fixed positive power gain.

The divisor estimates behind this history have different scopes. Tenenbaum I's main theorem uses positive-degree polynomials taking positive values on the positive integers, as fixed on printed p. 411 of the author-corrected reprint. Its Complement on p. 408 extends the lower bound in (1.13) and both bounds in (1.14) to every fixed $c_0<1$, after changing constants; the upper bound in (1.13) is not part of that extension. For irreducible $f$ in that standing class, the specialization (1.15)--(1.16) states

$$H_f(x,y,2y)=x(\log y)^{-\delta+o(1)}, \qquad \delta=1-\frac{1+\log\log2}{\log2},$$

as $x,y\to\infty$ with $y\leq x^{c_0}$, for a fixed $c_0<1$. Here $H_f$ counts inputs whose values have a divisor in $(y,2y]$. Paper I expressly does not reach $y$ of order $x$. Paper II, Theorem 1 supplies a different divisor lower bound for $y\leq x/2$. The library page's positive-degree restriction is editorial, excluding constant prime polynomials; Theorem 2 separately requires degree greater than one. These statements are not interchangeable fixed-power theorems.

Special polynomials

Earlier refereed results for single families come in two forms. Those proved for a positive proportion of a dyadic interval bound $F_f(n)$ at every large $n$ and have claim pages: Heath-Brown, Proc. London Math. Soc. 82 (2001), 554--596, for $t^3+2$ with exponent $1+10^{-303}$ (claim page); Irving, Acta Arith. 171 (2015), 67--80, for $t^3+2$ with exponent $1+10^{-52}$ (claim page); and de la Bretèche, Acta Arith. 169 (2015), 221--250, for even monic irreducible quartics with Galois group $V_4$ (claim page). The others state only that $P^+(f(m))$ exceeds a power $m^{1+c}$ for infinitely many $m$, which bounds $F_f(n)$ only along a sequence of $n$, so they have no claim page: Hooley, Acta Math. 117 (1967), 281--299, for $t^2+1$ with exponent $11/10$; Deshouillers--Iwaniec, Ann. Inst. Fourier 32 (1982), no. 4, 1--11, exponent about $1.2025$; de la Bretèche--Drappeau, J. Eur. Math. Soc. 22 (2020), 1577--1624, exponent $1.2182$; Merikoski, J. Eur. Math. Soc. 25 (2023), 1253--1284, exponent $1.279$; and Dartyge, Proc. London Math. Soc. 111 (2015), 1--62, for $t^4-t^2+1$. Merikoski's introduction and the introduction of Dartyge and Maynard's paper state these results in that form.

For every monic irreducible cubic, the unnumbered main theorem of Ermoshin (arXiv:2602.03642v3, 12 June 2026, p. 3) states that a positive proportion of $m\in[x,2x]$ have a prime factor of $f(m)$ exceeding $x^{1+c_f}$, for some $c_f>0$. It explicitly gives $F_f(n)\gg_f n^{1+c_f}$. This is an unconditional preprint result for monic irreducible cubics; neither a uniform exponent across polynomials nor the degree-three bound is supplied. No journal acceptance is recorded.

Dartyge and Maynard give a corresponding positive-proportion result for monic irreducible quartics whose Galois group is $C_4$ or $D_4$. Theorem 1.1 (arXiv:2212.03381v1, p. 3) states, for $x>x_0(f)$, that $\gg_f x$ integers $x<m\leq2x$ have $P^+(f(m))\geq x^{1+c_f}$, with $c_f>0$. Taking $x=n/2$ gives the elementary consequence $F_f(n)\gg_f n^{1+c_f}$ for this class. The publisher record confirms acceptance on 12 October 2023 and online publication in JEMS on 31 January 2025, DOI 10.4171/JEMS/1586. The statement locator above is to the arXiv preprint. This does not cover all quartics or give $n^4$.

For $f(t)=t^2+1$, Pascadi's published version of record (Forum of Mathematics, Pi 14 (2026), e8) has two relevant statements. Theorem 1.1 on p. 3 gives $P^+(m^2+1)>m^{1.3}$ for infinitely many individual $m$. The unnumbered assertion inside its proof instead gives

$$P^+\!\left(\prod_{x\leq m\leq2x}(m^2+1)\right)\geq x^{1.30008}$$

for every sufficiently large real $x$. Its locators are Section 6.3, Notation 6.7 on p. 49, (6.20) on p. 50, and the concluding estimates on pp. 51--52. Setting $x=n/2$ makes this dyadic product divide the initial product, so for all sufficiently large integers $n$ this consequence holds:

$$F_{t^2+1}(n)\geq2^{-1.30008}n^{1.30008}.$$

The factor $2^{-1.30008}$ belongs in the explicit inequality. This consequence is not Pascadi's theorem wording. The exact endpoint is the paper's numerical assertion.

A stronger quadratic preprint is Grimmelt--Merikoski, arXiv:2505.00493v2, 30 May 2025. Theorem 1.1 and its following paragraph (p. 2), specialized to $a=h=1$, give an $m\in[X,2X]$ with $P^+(m^2+1)>X^{1.312}$ for every sufficiently large $X$. The theorem's prime-sum hypothesis is stated there to hold unconditionally when $ah\leq X^{\varepsilon^2}$, which covers this specialization. Thus the same elementary interval inclusion gives

$$F_{t^2+1}(n)>2^{-1.312}n^{1.312}$$

eventually. This is the strongest exponent found for this polynomial in a preprint statement; no journal acceptance is recorded. Neither quadratic result gives the degree-two bound, an all-individual-$m$ power bound, or a result for every irreducible polynomial.

Adjacent results and a conditional lead

Pasten's arXiv:2609.01327v1 Theorem 1.1 and Corollary 1.2 (p. 1) concern each sufficiently large individual value $m^2+1$, including $P^+(m^2+1)\gg(\log_2m)^2/\log_4m$. Cuevas Barrientos--Pasten's arXiv:2504.15971v3 Theorem 1.3 (p. 2) gives related pointwise radical and prime-factor bounds for its quadratic and specified cubic families. Its phrase "two complex roots" is interpreted locally as distinct roots, not necessarily nonreal, using the discussion on p. 3 and the nonzero-discriminant construction on p. 8; "distinct" is not printed in the theorem line. These iterated-log bounds do not establish either requested power scale for the running product.

Bhalla's unpublished conditional note, studied in the conditional-route lead, records a route to $F_f(n)\gg_f n^d$ under its Hypothesis 3.1: for every irreducible $g\in\mathbb Z[x]$ with positive leading coefficient and no fixed prime divisor, there exist $A_g>1$ and $X_0(g)$ such that every real $X\geq X_0(g)$ admits an integer $t\in[X,A_gX]$ for which $g(t)$ is prime. Theorem 3.2 on p. 3 assumes that premise. The announcing post in the site's thread (16 April 2026) says the note was produced using GPT 5.4. The premise is unproved and the note's conditional proof is unreviewed; this is a research lead with no unconditional solution credit. The result is recorded as a conditional claim, pending and settling nothing unconditionally; the lead keeps its scope.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• bhalla_2026_conditional_note_large_prime_factors_polynomial_products
• cuevas_barrientos_2025_greatest_prime_factor_polynomial_values_subexponential_szpiro_families
• cuevas_barrientos_2025_greatest_prime_factor_polynomial_values_subexponential_szpiro_families / theorem_1_3
• dartyge_maynard_2025_largest_prime_factor_quartic_polynomial_values_cyclic_dihedral
• dartyge_maynard_2025_largest_prime_factor_quartic_polynomial_values_cyclic_dihedral / theorem_1_1
• erdos_1952_greatest_prime_factor
• erdos_1952_greatest_prime_factor / conjecture_p380
• erdos_1952_greatest_prime_factor / theorem
• erdos_1952_greatest_prime_factor / unproved_display_3
• erdos_1990_greatest_prime_factor
• ermoshin_2026_largest_prime_factor_irreducible_cubic_polynomial
• ermoshin_2026_largest_prime_factor_irreducible_cubic_polynomial / main_theorem
• grimmelt_merikoski_2025_greatest_prime_factor_uniform_equidistribution_quadratic_polynomials
• grimmelt_merikoski_2025_greatest_prime_factor_uniform_equidistribution_quadratic_polynomials / theorem_1_1
• pascadi_2026_large_sieve_exceptional_maass_greatest_prime_factor
• pascadi_2026_large_sieve_exceptional_maass_greatest_prime_factor / dyadic_product_bound
• pascadi_2026_large_sieve_exceptional_maass_greatest_prime_factor / theorem_1_1
• pascadi_2026_large_sieve_exceptional_maass_greatest_prime_factor / theorem_1_5
• pascadi_2026_large_sieve_exceptional_maass_greatest_prime_factor / theorem_1_7
• pasten_2026_improvement_largest_prime_factor_n_squared_plus_one
• pasten_2026_improvement_largest_prime_factor_n_squared_plus_one / corollary_1_2
• pasten_2026_improvement_largest_prime_factor_n_squared_plus_one / corollary_1_3
• pasten_2026_improvement_largest_prime_factor_n_squared_plus_one / theorem_1_1
• tenenbaum_1990_sur_une_question_erdos_schinzel_i
• tenenbaum_1990_sur_une_question_erdos_schinzel_i / irreducible_interval_corollary
• tenenbaum_1990_sur_une_question_erdos_schinzel_i / main_theorem
• tenenbaum_1990_sur_une_question_erdos_schinzel_ii
• tenenbaum_1990_sur_une_question_erdos_schinzel_ii / theorem_1
• tenenbaum_1990_sur_une_question_erdos_schinzel_ii / theorem_2
• tenenbaum_1990_sur_une_question_erdos_schinzel_ii / theorem_3
• erdos_1965_recent_advances_current_problems_number_theory