Denote by $P^+(n)$ (resp. $P^−(n)$) the largest (resp. the smallest) prime factor of the integer $n$. In this paper, we prove that there exists a positive proportion of integers $n$ having no small prime factor such that $P^+(n) x^{1/3−δ}, 0 < δ \leq 1/12$, where $P_3$ denote the integer having at most three prime factors taken with multiplicity. We also prove that the pattern $P^+(p − 1) < P^+(p + 1)$ holds for a positive proportion of primes under the Elliott-Halberstam conjecture.

https://hal.archives-ouvertes.fr/hal-01647898/document

Autour des plus grands facteurs premiers d’entiers consÉcutifs voisins d’un entier criblÉ

The cap set problem asks how large can a subset of \n\n \n \n (\n \n Z\n \n \n /\n \n 3\n \n Z\n \n \n )\n n\n \n \n (\\mathbb {Z}/3\\mathbb {Z})^n\n \n\n be and contain no lines or, more generally, how can large a subset of \n\n \n \n (\n \n Z\n \n \n /\n \n p\n \n Z\n \n \n )\n n\n \n \n (\\mathbb {Z}/p\\mathbb {Z})^n\n \n\n be and contain no arithmetic progressions. This problem was motivated by deep questions about structure in the prime numbers, the geometry of lattice points, and the design of statistical experiments. In 2016, Croot, Lev, and Pach solved the analogous problem in \n\n \n \n (\n \n Z\n \n \n /\n \n 4\n \n Z\n \n \n )\n n\n \n \n (\\mathbb {Z}/4\\mathbb {Z})^n\n \n\n, showing that the largest set without arithmetic progressions had size at most \n\n \n \n c\n n\n \n c^n\n \n\n for some \n\n \n \n c\n >\n 4\n \n c > 4\n \n\n. Their proof was as elegant as it was unexpected, being a departure from the tried and true methods of Fourier analysis that had dominated the field for half a century. Shortly thereafter, Ellenberg and Gijswijt leveraged their method to resolve the original cap set problem. This expository article covers the history and motivation for the cap set problem and some of the many applications of the technique: from removing triangles from graphs, to rigidity of matrices, and to algorithms for matrix multiplication. The latter application turns out to give back to the original problem, sharpening our understanding of the techniques involved and of what is needed for showing that the current bounds are tight. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, and the integers modulo \n\n \n N\n N\n \n\n.

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New applications of the polynomial method: The cap set conjecture and beyond

Perhaps the title \"Ramanujan and the birth of Probabilistic Number Theory\" would have been more appropriate and personal, but since Ramanujan's work influenced me greatly in other subjects too, I decided on this somewhat immodest title. Perhaps I should start at the beginning and relate how I first found out about Ramanujan's existence. In March 1931 I found a simple proof of the following old and well-known II p < 4 nn p<n theorem of Tchebychev : \"Given any integer n, there is always a prime p such that n < p < 2n .\" My paper was not very well written. Kalmar rewrote my paper and said in the introduction that Ramanujan found a somewhat similar proof. In fact the two proofs were very similar ; my proof had perhaps the advantage of being more arithmetical. He asked me to look it up in the Collected Works of Ramanujan which I immediately read with great interest. I very much enjoyed the beautiful obituary of Hardy in this volume [23]. I am not competent to write about much of Ramanujan's work on identities and on the ti-function since I never was good at finding identities. So I will ignore this aspect of Ramanujan's work here and many of my colleagues who are much more competent to write about it than I will do so. I will therefore write about Ramanujan's work on partitions and on prime numbers and here too I will restrict myself to the asymptotic theory. My paper [7] on Tchebychev's theorem, which was actually my very first, appeared in 1932. One of the key lemmas was the proof that (1)

Ramanujan and I

Take the product of the numbers (n/(n+1))∊n for 1≤ n < N, where each ∊n is ± 1. Express the product as a/b in lowest terms. Evidently the minimal possible value for a over all choices for ∊n is 1; just take each ∊n = 1, or each ∊n = 0. Denote the maximal possible value of a by A(N). It is known from work of Nicolas and Langevin that (log 4+o(1))N≤ log A(N)≤(2/3+o(1))Nlog N. Using the Rosse–Iwaniec sieve, we improve the lower bound to the same order of magnitude as the upper bound.

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Products of Ratios of Consecutive Integers

There are many ways to recall Paul Erdős’ memory and his special way of doing mathematics. Ernst Straus described him as “the prince of problem solvers and the absolute monarch of problem posers”. Indeed, those mathematicians who are old enough to have attended some of his lectures will remember that, after his talks, chairmen used to slightly depart from standard conduct, not asking if there were any questions but if there were any answers.

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Some of Erdős’ Unconventional Problems in Number Theory, Thirty-four Years Later

is less than Ex . Our proof entails Brun's method . A corollary is that the probability that P(n) > P(n + 1) is positive (almost certainly this probability is 1/2) . Our methods allow us to say something about integers n which have the same sum of their prime factors as n + 1 . We prove the number of such n :5 x is O(x/(log x)' -') for every E > 0. We know a proof in the case E = 0 as well, but it is more complicated and not presented . In addition we present a brief discussion on the largest prime factors of 3 or more consecutive numbers .

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Paul Erdös

Papers

On the largest prime factors of n and n + 1