By Claudi Alsina, Roger B. Nelsen

Theorems and their proofs lie on the center of arithmetic. In talking of the merely aesthetic features 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'. fascinating Proofs offers a set of outstanding proofs in uncomplicated arithmetic which are enormously stylish, jam-packed with ingenuity, and succinct. by way of a shocking argument or a robust visible illustration, the proofs during this assortment will invite readers to benefit from the great thing about arithmetic, and to strengthen the facility to create proofs themselves. The authors ponder proofs from issues akin to geometry, quantity idea, inequalities, aircraft tilings, origami and polyhedra. Secondary tuition and college lecturers can use this ebook to introduce their scholars to mathematical splendor. greater than one hundred thirty workouts for the reader (with options) also are integrated.

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2 Sums of squares, triangular numbers, and cubes Having examined triangular numbers and squares as sums of integers and sums of odd integers, we now consider sums of triangular numbers and sums of squares. 7. 2n C 1/ . 6 Proof. We give two proofs. n C 1/=2 [Gardner, 1973]. 9. 9. 2. 1 C 2 C C n/ from which the result follows. In the second proof, we write each square k 2 as a sum of k ks, then place those numbers in a triangular array, create two more arrays by rotating the triangular array by 120ı and 240ı , and add corresponding entries in each triangular array.

1) Now F . k/ . 0/ are integers. x/ is f . x/ sin x Ä f . 1) is positive but arbitrarily close to 0 for n sufficiently large. 1) is false, as is our assumption that is rational. ✐ ✐ ✐ ✐ ✐ ✐ “MABK014-02” — 2010/7/10 — 13:30 — page 28 — #10 ✐ 28 ✐ CHAPTER 2. Distinguished Numbers The Biblical value of We have all heard that somewhere in the Bible it says that equals 3. The source of this claim is I Kings 7:23 (also II Chronicles 4:2), where the Bible (New International Version) describes a round “sea” (large bowl) made of metal, 10 cubits in diameter and 30 cubits in circumference.

Pierre de Fermat (1601–1665) mentioned it (without proof ) in a letter in 1640; Leonhard Euler published the first proof in 1736. There are many known proofs of Fermat’s theorem. The one we present is particularly elegant. 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.

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