An Invitation to Mathematics: From Competitions to Research by Dierk Schleicher, Malte Lackmann

By Dierk Schleicher, Malte Lackmann

This Invitation to arithmetic involves 14 contributions, many from the world's prime mathematicians, which introduce the readers to intriguing facets of present mathematical learn. The contributions are as diversified because the personalities of lively mathematicians, yet jointly they convey arithmetic as a wealthy and vigorous box of study. The contributions are written for scholars on the age of transition among highschool and collage who understand highschool arithmetic and maybe festival arithmetic and who are looking to discover what present learn arithmetic is set. we are hoping that it'll even be of curiosity to lecturers or extra complex mathematicians who want to find out about interesting facets of arithmetic outdoor in their personal paintings or specialization. including a staff of younger ``test readers'', editors and authors have taken nice care, via a considerable ``active editing'' approach, to make the contributions comprehensible through the meant readership.

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However any mathematician who poses the problem in this form is likely to be seen as being divorced from reality, so let us switch from sex to socks and pose the following problem: Problem 2. A man has a selection of socks of three different colours, which he keeps in a bag. Two socks of the same colour may be assumed to form a pair. He finds that if he pulls out two socks at random then there is exactly an even chance that they will form a pair. What can be said about how many socks of each colour he has?

By the analogue of Theorem 4, we see that 2 no number between 2 and 40 divides a number of the form d +163 . 4 The rest is straightforward: Assume that x2 + x + 41 is not prime for some −40 ≤ x ≤ 39. Then it has a prime factor p with 1

The next example after that is when Δ = 229, generated for example by (−9, 3, 5). Again there is a unique self-reflecting circuit, and as 229 is square-free this gives the first example of a quadratic field whose class number, 3, is not a power of 2. • The first Δ with more than 2 primitive self-reflecting circuits is 480. From Sex to Quadratic Forms 41 11 Conclusion Our journey, which started with an elementary problem, has taken us through several different branches of mathematics, thus illustrating the unity of the subject.

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