Every-interval smooth-number theorems matched at the diagonal

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This page supports the proof sketch for upper bounds on k(n). Problem 962 needs, for every block $(m,m+K]$ with $m\le n$, a member whose prime factors are all at most $K$: the interval length and the smoothness bound are the same number. The comparison here applies a theorem with interval length $x^\beta$ and smoothness $x^\alpha$ by scaling with a multiplier. The sketch shows that this application never goes below $\beta$ and gives $\alpha/(1+\alpha-\beta)$ when $\beta\le\alpha$. This limits that application, not every possible use of the theorem. Each entry records what was read and how deeply.

Theorems that bear on the diagonal

Erdős and Sárközy (1979). P. Erdős and A. Sárközy, On the prime factors of $\binom nk$ and of consecutive integers, Utilitas Math. 16 (1979), 197--215; public scan in the Rényi Institute's Erdős archive. Theorem 2 (printed p. 198): $n_k>k^{5/2}/16$ for $k>k_0$, with $n_k$ the smallest $n$ such that $P(n+i)>k$ for $1\le i\le k$ (printed p. 197). The definition and the statement were checked on page images of the scan; the proof in section 3 was read in the scan's text layer. The construction is the one-multiplier near-square adjustment, and the theorem gives the exponent $2/5$ for $k(n)$; L250 restates it with explicit constants.

Friedlander and Lagarias (1987). J. B. Friedlander and J. C. Lagarias, On the distribution in short intervals of integers having no large prime factor, J. Number Theory 25 (1987), 249--273. Read in the authors' manuscript: the definition of $f(\alpha)$, the infimum of the $\beta$ such that, for every $\alpha_1>\alpha$, every interval $(x-x^\beta,x]$ with sufficiently large $x$ contains an $x^{\alpha_1}$-smooth integer; Theorem 3, $f(\lambda\alpha)\le\lambda f(\alpha)+1-\lambda$; Theorem 4, $f(1/r)\le1-2/r+1/(r2^{r-1})$, with the case $r=2$ credited also to Balog and Sárközy. Statements, with the proof of Theorem 3 and the start of the proof of Theorem 4. Diagonal: $f(1/2)\le1/4$ gives $2/5$ (with an $\varepsilon$ of slack); $f(1/3)\le5/12$ gives $5/12$; for $r\ge4$ the bound of Theorem 4 exceeds $1/2$. Translating the endpoint and allowing arbitrarily small power slack gives the forward-interval form used for the comparison.

Harman (1999). G. Harman, Integers without large prime factors in short intervals and arithmetic progressions, Acta Arith. 91 (1999), 279--289. Theorem 2 (printed p. 280), read on the public archive scan at statement level: for $y=x^\gamma$ with $3/7<\gamma<1/2$, with $\theta(\gamma)$ an infimum over known exponent pairs, if $\theta(\gamma)<\gamma$ then $\Psi(x+y,u)-\Psi(x,u)\gg y$ for $u\ge x^{\varrho\theta+\varepsilon}$, $\varrho=e^{-1/2}$. The value $\theta=(3-5\gamma)/2$ on $3/7<\gamma\le1/2$ is conditional on the exponent pair conjecture; with the pairs known in 1999 the author remarks that the result holds down to some $\gamma$ between $0.45$ and $0.46$. Since the smoothness exponent $\varrho\theta(\gamma)$ is below $\gamma$, scaling gives a diagonal exponent of at least $\gamma$: above $0.45$ unconditionally, above $3/7$ under the conjecture. The $0.45$ comparison uses the pairs in the author's 1999 discussion; the theorem's restriction $\gamma>3/7$ itself already prevents an improvement of $2/5$ through scaling. For a concrete further match, inserting Bourgain's pair from L253 into Harman's formula (1.2) makes $\theta(\gamma)<\gamma$ whenever $\gamma>(139+84\eta)/(307+84\eta)$, so it supplies every fixed interval exponent above $139/307\approx0.4528$ by taking $\eta$ small enough. This is an application calculated here, not a claim that the infimum over all available pairs equals that value. Granville's survey (2008, section 4.1, printed p. 295) reports the range $1/2\ge\beta\ge3/7$ with $\alpha>(3-5\beta)/(2\sqrt e)$ without stating this condition; the original's hypotheses govern.

Weingartner (2023). Somewhat smooth numbers in short intervals, Ramanujan J. 60 (2023), 447--453, read as arXiv:2105.13568v2, Theorem 1, Corollaries 1--3 and Table 1 at statement depth, with the opening divisor count in the proof of Theorem 1. This is a uniform result with smoothness $y\ge\sqrt{2x}$ and length

$$z\ge A\min\left(x^{\vartheta}, x^{\ell/(1+\kappa)}y^{(\kappa-\ell)/(1+\kappa)}\right),$$

where $(\kappa,\ell)$ is an exponent pair and the reported value of $\vartheta$ is $517/1648+\varepsilon>1/4$. At $y=x^\alpha$ with $\alpha\ge1/2$, the second interval exponent is $\beta=((1-\alpha)\ell+\alpha\kappa)/(1+\kappa)$. Since $\ell\ge1/2$ and $\kappa\ge0$, it obeys $\beta\ge(1-\alpha)/2$; the first exponent $\vartheta$ also obeys that inequality. Scaling therefore gives a diagonal exponent at least $2\alpha/(1+3\alpha)\ge2/5$. This comparison covers the whole stated exponent-pair family, even its conjectured limit. The paper also states the near-square single-witness bound with smoothness $\sqrt{2x}$ and length $3x^{1/4}$, which scales to the classical $2/5$.

Bourgain (2017), an analytic input. Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), 205--224; arXiv:1408.5794v2, Section 5, Theorem 6 and its proof discussion read in the primary text. It proves that $(13/84+\eta,55/84+\eta)$ is an exponent pair for every $\eta>0$, including the monomial phase and parameter range used in L253's derivation of the diagonal bound 38/97+epsilon. This is a project application of the source theorem, not a smooth-block theorem stated by Bourgain. The full decoupling proof is an external dependency and was not independently verified here.

Results at the square-root scale or longer

These give no exponent below $1/2$ by scaling, because their interval exponent is at least $1/2$.

• Balog, On the distribution of integers having no large prime factor, Astérisque 147--148 (1987), 27--31: the statement and its deduction on printed pp. 27--28 (text layer, partly garbled): smoothness $x^\alpha$ for every fixed $\alpha>0$ in intervals of length $x^{1/2+\varepsilon}$.
• Harman (1999), Theorem 1 (printed p. 279): intervals $[x,x+x^{1/2}]$, smoothness $x^{\varrho/4+\varepsilon}$.
• Croot, Smooth numbers in short intervals (author manuscript, 2006), Theorem 1: intervals of length $c(\varepsilon)\sqrt x$, smoothness $x^{47/(190\sqrt e)+\varepsilon}$; statement only. Granville cites a different version with a different constant.
• Matomäki and Radziwiłł (2016), filed in the library: its Corollary 1 gives X^epsilon-smooth numbers in every interval of length C(epsilon) sqrt(X), at the square-root scale, and its Corollary 6 counts smooth numbers only in almost all short intervals, so it cannot control an exceptional block.
• Jain, Smooth numbers in short intervals, arXiv:2502.10530v1, Theorems 1.1 and 1.2 (statements): Theorem 1.2 holds in every interval but needs length at least $\sqrt x\,e^{(1+\varepsilon)W}$, where $W=\frac{11}{16}u\log u+2\log\log x$ and $u=\log x/\log y$; Theorem 1.1 is an almost-all result. A 2026 journal version (Quart. J. Math. 77, 397--422) exists; its final statements were not read.
• Younis, Asymptotics for smooth numbers in short intervals, arXiv:2409.05761v1, Theorem 1.1 (statement): lengths $x^\theta$ with $\theta>17/30$.

Almost-all theorems (Friedlander and Lagarias's Theorem 6, Hildebrand--Tenenbaum, Jain's Theorem 1.1, Matomäki and Radziwiłł's Corollary 6) do not apply to $k(n)$ at all: the one bad block may lie in the exceptional set.

Related methods and different problems

• Rivat and Shparlinski, Multiples of squares in short intervals, Funct. Approx. Comment. Math. 54 (2016), 57--63: by its abstract, exponent pairs and Vaaler polynomials give a multiple $m^2r$ of a square in every interval $[x,x+x^{1/2}]$ with $m>x^{0.281286}$. It is the closest method precedent for the multiplier sum of L251, in a different range: there the interval has length $x^{1/2}$ and the square root $m$ should be large, while L251 needs $cq^2$ in windows of length about $x^{4/7}$ with $q\asymp x^{11/28}$ capped by $K$ and the multiplier in a dyadic range. The full text was not accessible; only the abstract was read.
• van Doorn and Tang, Consecutive integers free of certain prime factors, arXiv:2606.19863 (abstract): their $n_k$ is the least $n>2k$ such that $(n-k)\cdots(n-1)$ has no prime factor in $(k,2k)$, a different function; their superpolynomial lower bound says nothing about $k(n)$ of Problem 962.
• An author report on Problem 962 at erdosproblemaday.com/report/962 (accessed 2026-10-15) claims $k(n)<n^{0.46}$ through a theorem of Laishram and Murty, Grimm's conjecture and smooth numbers, Michigan Math. J. 61 (2012), 151--160. Their abstract (arXiv:1306.0765) proves $g(n)=O(n^\alpha)$ with $0.45<\alpha<0.46$ for Grimm's function $g(n)$, the same range that Harman's Theorem 2 reaches with known exponent pairs; the paper itself was not read, and the exponent is weaker than $2/5$ in any case.

Search record

A search on 2026-10-15 for every-interval results with a diagonal exponent below $2/5$, or for improvements of $n_k>k^{5/2}/16$, found none. It covered web searches on the phrases "every interval" smooth exponent below $2/5$, "maximal gaps between smooth numbers", "Erdős problem 962" with $n^{2/5}$, "$n_k$" with $k^{5/2}$ and with "prime factor greater than $k$", and multiples of squares or almost squares in short intervals with the exponents $11/28$ and $38/97$; the zbMATH review of the 1979 paper (Zbl 0419.10040); a zbMATH query for documents whose reference lists contain that paper (no records returned, which is weak evidence since older reference lists are often incomplete); the arXiv listing API; and the EuDML record of Harman's paper. The sources above were then read at the depths stated. This is a bounded search, not a novelty certificate: MathSciNet and Google Scholar were not searched, and Balog and Sárközy's papers on sums of integers with small prime factors (Studia Sci. Math. Hungar. 19 (1984), 35--47 and 81--88), which Friedlander and Lagarias credit for the case $r=2$, were not read.