{"title":"4. Congruences, clocks, and calendars","authors":"Robin Wilson","doi":"10.1093/actrade/9780198798095.003.0004","DOIUrl":"https://doi.org/10.1093/actrade/9780198798095.003.0004","url":null,"abstract":"‘Congruences, clocks, and calendars’ demonstrates how we might apply the idea of congruence, first introduced by Gauss in 1801, to problems such as testing which Mersenne numbers are primes and finding the day of the week on which a given date falls. Ancient Chinese puzzles depended on the solving of simultaneous linear congruences, inspiring mathematicians and giving rise to the Chinese Remainder Theorem. Exploring quadratic congruences leads towards the law of quadratic reciprocity, noted by Euler and Legendre and proved by Gauss. The problem, ‘Is 1066 a square or a non-square?’ can be solved by applying this law several times to reduce the numbers involved.","PeriodicalId":190248,"journal":{"name":"Number Theory: A Very Short Introduction","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124323955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"8. How to win a million dollars","authors":"Robin Wilson","doi":"10.1093/actrade/9780198798095.003.0008","DOIUrl":"https://doi.org/10.1093/actrade/9780198798095.003.0008","url":null,"abstract":"What is the Riemann hypothesis, and why does it matter? ‘How to win a million dollars’ looks in detail at Riemann’s conjecture. While Gauss attempted to explain why primes thin out, Bernhard Riemann in 1859 proposed an exact formula for the distribution of primes, employing Euler’s ‘zeta function’ and the idea of complex numbers. In 2000, the Clay Mathematics Institute offered a million dollars for the solutions of each of seven famous problems, of which the Riemann hypothesis was one. The Riemann hypothesis implies strong bounds on the growth of other arithmetic functions, in addition to the primes-counting function. It remains one of the most famous unsolved problems of mathematics.","PeriodicalId":190248,"journal":{"name":"Number Theory: A Very Short Introduction","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129986831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}