Largest prime factors of consecutive integers: a joint Dickman law

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Family 012 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

The integers $n$ and $n+1$ share no prime factors. Does this local restriction leave a lasting connection between the sizes of their largest prime factors?

For example, sixteen has largest prime factor two, while seventeen is itself prime. Other neighboring pairs have the opposite ordering. A longstanding prediction says that each ordering should occur half the time. A stronger conjecture predicts a complete joint distribution: after taking logarithms and scaling by the logarithm of the integer, the two largest factors should behave like independent samples from the classical Dickman distribution.

Family 012 claims this stronger result in ordinary natural density. Every integer up to the cutoff is weighted equally, and the limit is asserted through every growing cutoff. The familiar half-and-half ordering is then a consequence of the joint law, rather than a separate analytic argument.

The challenge is dependence. Knowing the distribution of the largest factor of one integer does not determine the distribution of a neighboring pair, just as knowing two marginal probability distributions does not establish independence.

2. Background and essential definitions

Write $P^+(n)$ for the largest prime factor of $n>1$, and put $P^+(1)=1$. An integer is $y$-smooth when all its prime factors are at most $y$. Thus the condition

$$P^+(n)\le n^a$$

asks whether every prime factor is below a fixed power of the integer, with $0<a<1$.

The natural normalization is

$$U_n=\frac{\log P^+(n)}{\log n}, \qquad V_n=\frac{\log P^+(n+1)}{\log n}.$$

Both use the denominator $\log n$, as in the manuscript. Replacing the second denominator by $\log(n+1)$ would have the same limit, but keeping one convention makes the ordering comparison exact.

For a set $A$ of positive integers, natural density is the limit

$$\lim_{X\to\infty}\frac{\#(A\cap[1,X])}{X},$$

when it exists. A positive lower density only says that the lower limit is positive; it need not imply convergence. Logarithmic density instead uses weights $1/n$ and normalization $\log X$. It averages information across multiplicative scales and is weaker for this purpose than a conclusion at every ordinary cutoff.

The Dickman–de Bruijn function $\rho$ is defined by

$$\rho(u)=1\quad(0\le u\le1), \qquad u\rho'(u)=-\rho(u-1)\quad(u>1).$$

The equation determines successive intervals from earlier ones. For example, on $1\le u\le2$, it gives $\rho(u)=1-\log u$. Classical smooth-number theory says that the limiting distribution function of $U_n$ is

$$H(a)=\rho(1/a),\qquad 0<a\le1.$$

It extends continuously to $[0,1]$. The proposed independence statement says that the probability of two threshold conditions tends to the product $H(a)H(b)$.

This does not mean that all divisibility events at consecutive integers are independent. A prime cannot divide both $n$ and $n+1$. The claim concerns a particular large-scale statistic after normalization, not the full factorization data.

3. Development history and previous partial results

Dickman’s 1930 work and later results of Ramaswami and de Bruijn established the one-variable smoothness law. That marginal distribution is an input and historical starting point here, not the manuscript’s new conclusion.

Erdős and Pomerance’s 1978 paper formulated the joint problem and proved that each ordering has positive lower natural density, with an explicit bound of $0.0099$. They also showed that the ratio of the two largest factors rarely stays too close on suitable power scales. These results revealed substantial variation without identifying a limiting distribution.

Subsequent work raised the ordering lower bounds. Results of de la Bretèche–Pomerance–Tenenbaum, Fouvry, Wang, and Lü–Wang led to the value $0.2017$. Yang’s 2026 preprint gives $0.280$ for both orderings. In this progression, a lower bound should not be confused with existence of a natural density or the exact value one-half.

Teräväinen’s binary-correlation theorem proved the predicted joint law in logarithmic density, and hence logarithmic density one-half for the ordering. It also gave positive lower natural density for every nondegenerate interior rectangle of normalized values. This was much more than knowing that both orderings occur often, but still involved a different averaging conclusion.

Tao and Teräväinen obtained the ordinary joint law outside an exceptional set of scales of logarithmic density zero. Their later quantitative work provides logarithmic savings and parameter uniformity while retaining exceptional scales. Other work proved conditional ordinary laws under distribution assumptions for smooth numbers, or averaged the problem over shifts.

M1 claims an unconditional ordinary limit at the fixed shift one with no exceptional scales. It does not claim to retain the quantitative error term or growing-parameter uniformity of those other results.

4. What the new manuscripts claim

M1, Theorem 1.1, states that for every fixed $a,b\in(0,1)$,

$$\lim_{X\to\infty}\frac1X \#\left\{2\le n\le X: \begin{array}{l} P^+(n)\le n^a,\\ P^+(n+1)\le n^b \end{array}\right\} =\rho(1/a)\rho(1/b).$$

The limit is through all real $X$, with ordinary unweighted counting. The manuscript first obtains the corresponding theorem with fixed thresholds $X^a,X^b$, then passes to the displayed moving thresholds.

For a concrete consequence, taking $a=b=1/2$ gives the limiting proportion

$$\rho(2)^2=(1-\log2)^2,$$

approximately $9.4\%$. This is what the claimed theorem predicts for pairs in which both largest factors are at most $\sqrt n$; it is not a reported numerical experiment.

Combining the joint threshold law with the marginals also gives independence for upper-tail events, recovering the Erdős–Pomerance formulation. More generally, the limiting joint distribution of $(U_n,V_n)$ is the product of two identical continuous Dickman distributions.

Corollary 1.2 asserts

$$\lim_{X\to\infty}\frac1X \#\{2\le n\le X:P^+(n)<P^+(n+1)\} =\frac12.$$

The reverse ordering has the same density. The deduction uses the symmetry and continuity of the limiting product distribution: it assigns equal mass to the two sides of the diagonal and zero mass to the diagonal itself. No symmetry of the finite list of consecutive pairs is assumed.

5. Why the advance matters

Provisional importance: 4/5. The claim would resolve a longstanding independence problem for consecutive integers in the original ordinary-density form. It identifies an entire joint law, of which the ordering question is one particularly accessible consequence.

The result would also demonstrate how a strong local dependence can disappear from a normalized large-factor statistic. Consecutive integers remain coprime, but their limiting largest-factor sizes would nevertheless be independent. Such statements are precise versions of probabilistic intuition in number theory.

Removing exceptional scales is mathematically meaningful. A theorem valid at almost all scales may still leave arbitrarily large cutoffs with different behavior. M1 claims to exclude every such bad sequence for fixed smoothness parameters. The rating reflects this advance and the proposed methods if correct, without reading a quantitative convergence speed into a qualitative limit.

6. The main idea

Encode the largest factor with finitely many labels

At a large scale $x$, divide primes above $x^{1/J}$ into finitely many logarithmic bins,

$$\bigl(x^{k/J},x^{(k+1)/J}\bigr], \qquad 1\le k<J,$$

with $J$ fixed. Counting prime factors in these bins gives a finite description of the large-factor pattern. The counts are bounded on the relevant integer range because too many large factors would make their product exceed that range.

Assign a unit complex number $\zeta_k$ to each bin and encode its count by a power of $\zeta_k$. Multiplying these powers produces a multiplicative bin function. Finite Fourier inversion recovers the probabilities of count patterns from averages of such functions, so independence reduces to decorrelation for arbitrary pairs of phase choices.

Subtract the mean from one bin function to obtain a centered function $F_x$, and call the other $g_x$. The required statement has the shape

$$\frac1x\sum_{n<x}g_x(n)F_x(n+1)\longrightarrow0.$$

The centered function is no longer multiplicative. This prevents a direct substitution into a general theorem about multiplicative functions. The manuscript instead exploits that multiplication by any fixed integer changes none of the large-prime bins once $x$ is sufficiently large.

Amplify a hypothetical correlation

Assume a nonzero correlation persists along some sequence of scales. M1 introduces nonnegative divisor weights built at an auxiliary scale $B$. These weights amplify the putative bias while having controlled first and second moments.

Writing the integer through the chosen divisors permits the fixed-multiplier invariance to simplify one endpoint. Cauchy–Schwarz removes the bounded function $g_x$ and yields a quadratic energy involving two copies of $F_x$. After discarding a negligible diagonal, an exact change of variables converts the remaining terms into shifted products

$$F_x(n')\,\overline{F_x(n'+j)}.$$

The result is a weighted graph of ordinary additive shifts. A persistent original correlation forces a positive amount of energy in this graph.

Approximate the graph by separate endpoint features

The edge weights depend on auxiliary-prime divisibility at both endpoints and on residue information. M1 compares these data with independent random prime sets, where each auxiliary prime is included with probability $1/p$.

The goal is not to declare the actual integers independent. It is to approximate the specific graph kernel well enough for every bounded pairing that appears in the argument. The relevant cut norm measures the largest normalized matrix pairing with bounded test vectors, allowing those vectors to depend on the realized matrix.

Randomly splitting each endpoint’s prime set smooths its logarithmic and residue information. The manuscript uses this to approximate the kernel by a fixed finite sum of products of one-endpoint features. A sampling argument upgrades an integrated comparison to the needed cut-norm comparison; second-moment bounds control the potentially large kernel entries.

The features can then be approximated by expressions of the form

$$G_{B,v}(n)= \prod_{\substack{p\in\mathcal P_B\\p\mid n}} \frac{1+p^{-v/B}}2, \qquad v\ge0,$$

where $\mathcal P_B$ is the chosen auxiliary prime set. This explicit form links the graph estimate back to short averages of arithmetic functions.

Apply short-interval cancellation

The manuscript establishes cancellation for short averages of $F_xG_{B,v}$, including fixed residue restrictions, using the short-interval theorems of Matomäki–Radziwiłł and Matomäki–Radziwiłł–Tao. Because the bin counts lie in a finite range, interpolation by nonnegative multiplicative functions handles the fact that $F_x$ itself is centered rather than multiplicative.

After the graph kernel is separated into endpoint features, its energy reduces to products of those short averages. Their vanishing contradicts the positive energy forced by the hypothetical bias.

The order of limits is part of the mechanism: the original scale $x$ goes to infinity first, with auxiliary data fixed, and then $B$ grows. The contradiction is applied to a subsequence of any proposed bad sequence of cutoffs, producing the claimed full ordinary limit. Refining the finite bins then recovers the largest-factor distribution.

7. Scope, limitations, and open questions

The theorem fixes $a$ and $b$ in $(0,1)$. It gives no stated error rate and no uniformity when the thresholds vary with $X$. The comparison with earlier quantitative work should preserve that tradeoff: removing exceptional scales does not automatically retain stronger rates.

The displayed result concerns $n$ and $n+1$. It does not state an arbitrary-length independence theorem for consecutive blocks or the entire joint factorization process. It also differs from family 011, which samples primes and studies the ranked factors of their predecessors.

The largest-factor statistic cannot encode all small-prime correlations, parity information, or coprimality constraints. Those features remain present even if this normalized joint law holds. Extensions to more shifts, sharper error bounds, and varying smoothness parameters are further questions beyond the stated conclusion.

8. Sources and further reading

M1’s Theorem 1.1 and Corollary 1.2 give the joint law and ordering density. Its introduction describes the bin encoding, divisor amplification, graph comparison, and limit order. Section 11 connects the resulting correlations to the moving-threshold law and the ordering consequence.

The primary sources in Section 3 distinguish marginal smoothness, logarithmic density, lower natural density, and almost-all-scale results. For optional definitions, see smooth numbers, natural density, and the Dickman function.

Source manuscripts

• M1. OpenAI, The joint Dickman law for consecutive integers (September 24, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.