{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11242","name":"Making Squares - Sieves, Smooth Numbers, Cores and Random Xorsat (Keynote Speakers)","abstract":"Since the advent of fast computers, much attention has been paid to practical factoring algorithms. Several of these algorithms set out to find two squares x^2, y^2 that are congruent modulo the number n we wish to factor, and are non-trivial in the sense that x is not equivalent to +\/- y mod n. In 1994, this prompted Pomerance to ask the following question.\nLet a_1, a_2, ... be random integers, chosen independently and uniformly from a set {1, ... x}. Let N be the smallest index such that {a_1, ... , a_N} contains a subsequence, the product of whose elements is a perfect square. What can you say about this random number N? In particular, give bounds N_0 and N_1 such that P(N_0 <= N <= N_1)-> 1 as x -> infty. Pomerance also gave bounds N_0 and N_1 with log N_0 ~ log N_1.\nIn 2012, Croot, Granville, Pemantle and Tetali significantly improved these bounds of Pomerance, bringing them within a constant of each other, and conjectured that their upper bound is sharp. In a recent paper, Paul Balister, Rob Morris and I have proved this conjecture. In the talk I shall review some related results and sketch some of the ideas used in our proof.","keywords":["integer factorization","perfect square","random graph process"],"author":{"@type":"Person","name":"Bollob\u00e1s, B\u00e9la","givenName":"B\u00e9la","familyName":"Bollob\u00e1s","affiliation":"University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK and University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152, USA"},"position":3,"pageStart":"3:1","pageEnd":"3:1","dateCreated":"2018-06-18","datePublished":"2018-06-18","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Bollob\u00e1s, B\u00e9la","givenName":"B\u00e9la","familyName":"Bollob\u00e1s","affiliation":"University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK and University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152, USA"},"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.AofA.2018.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6313","volumeNumber":110,"name":"29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)","dateCreated":"2018-06-18","datePublished":"2018-06-18","editor":[{"@type":"Person","name":"Fill, James Allen","givenName":"James Allen","familyName":"Fill"},{"@type":"Person","name":"Ward, Mark Daniel","givenName":"Mark Daniel","familyName":"Ward"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11242","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6313"}}}