By Claudi Alsina, Roger B. Nelsen

Theorems and their proofs lie on the middle of arithmetic. In talking of the simply aesthetic characteristics of theorems and proofs, G. H. Hardy wrote that during appealing proofs 'there is a truly excessive measure of unexpectedness, mixed with inevitability and economy'. captivating Proofs offers a suite of outstanding proofs in straight forward arithmetic which are really stylish, jam-packed with ingenuity, and succinct. by way of a shocking argument or a strong visible illustration, the proofs during this assortment will invite readers to benefit from the fantastic thing about arithmetic, and to improve the facility to create proofs themselves. The authors give some thought to proofs from themes resembling geometry, quantity thought, inequalities, aircraft tilings, origami and polyhedra. Secondary university and college academics can use this publication to introduce their scholars to mathematical attractiveness. greater than a hundred thirty workouts for the reader (with options) also are incorporated.

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The theorem is clearly true for p D 2, so assume p odd. From Fermat’s theorem, the integers 1, 2, . . mod p/. mod p/, as required. 1 Á . n 1/Š Á 1. mod n/, then n is prime. To prove this, assume n is not prime, so that n D ab with 1 < a; b < n 1. mod n/. 7 Perfect numbers Perfect numbers have engaged the attention of arithmeticians of every century of the Christian era. L. E. Dickson History of the Theory of Numbers Perfect numbers like perfect men are very rare. Ren´e Descartes A perfect number is a positive integer n that is equal to the sum of its positive divisors excluding itself.

1 Pick’s theorem Pick’s theorem is admired for its elegance and its simplicity; it is a gem of elementary geometry. Although it was first published in 1899, it did not attract much attention until seventy years later when Hugo Steinhaus included it in the first edition of his lovely book Mathematical Snapshots [Steinhaus, 1969]. Georg Alexander Pick (1859–1942) was born in Vienna but lived much of his life in Prague. Pick wrote many mathematical papers in the areas of differential equations, complex analysis, and differential geometry.

In it we simply count objects in a set [Golomb, 1956]. 17. If n is an integer and p a prime, then p divides np n. Furthermore, if n is not a multiple of p, then p divides np 1 1. Proof. 7). Suppose we have a supply of beads in n distinct colors, and we wish to make multicolored necklaces consisting of p beads. We first place p beads on a string. Since each bead can be chosen in n ways, there are np possible strings of beads. For each of the n colors, there is one string whose beads are all the same color, which we discard, leaving np n strings.