Download Number Theory: A Historical Approach by John J. Watkins PDF

By John J. Watkins

The common numbers were studied for hundreds of thousands of years, but such a lot undergraduate textbooks current quantity concept as a protracted checklist of theorems with little point out of ways those effects have been came upon or why they're very important. This booklet emphasizes the old improvement of quantity conception, describing equipment, theorems, and proofs within the contexts within which they originated, and offering an available advent to at least one of the main interesting topics in mathematics.

Written in an off-the-cuff type by way of an award-winning instructor, Number Theory covers leading numbers, Fibonacci numbers, and a number of different crucial subject matters in quantity conception, whereas additionally telling the tales of the nice mathematicians at the back of those advancements, together with Euclid, Carl Friedrich Gauss, and Sophie Germain. This special introductory textbook positive aspects an in depth set of difficulties that allow scholars to actively toughen and expand their realizing of the fabric, in addition to totally labored strategies for plenty of of those difficulties. it's also valuable tricks for whilst scholars are uncertain of the way to start on a given problem.

  • Uses a distinct historic method of instructing quantity theory
  • Features quite a few difficulties, necessary tricks, and entirely labored solutions
  • Discusses enjoyable subject matters like Pythagorean tuning in tune, Sudoku puzzles, and mathematics progressions of primes
  • Includes an creation to Sage, an easy-to-learn but robust open-source arithmetic software program package deal
  • Ideal for undergraduate arithmetic majors in addition to non-math majors
  • Digital recommendations guide (available in simple terms to professors)

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Extra resources for Number Theory: A Historical Approach

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100, is that eight times a triangular number plus one always yields a square. So, for example, 8 · 6 + 1 = 49, and 8 · 15 + 1 = 121 = 112 . This fact is easy to prove algebraically since we can write 8 · tn + 1 = 8n(n + 1) + 1 = 4n2 + 4n + 1 = (2n + 1)2 . 3. Tetrahedral and Pyramidal Numbers The geometrical properties of numbers studied in ancient times were not limited to two dimensions. Even in modern times, as you travel around the world you see fruit and produce stacked in geometric patterns in markets or by the roadside.

Then express the reciprocal of 7 both as a decimal fraction and as a sexagesimal fraction. 9 (S) Here are the values of s and t that correspond to the fifteen rows of Plimpton 322. ) 21 Number Theory Begins Row 1 2 3 4 5 6 7 8 9 10 12 13 14 15 s 12 64 75 125 9 20 54 32 25 81 48 15 50 9 t 5 5 27 32 54 4 9 25 15 12 40 25 8 27 Note that a few values seem to be missing such as s = 5, t = 2, which give triangle {21, 20, 29}. 60 degrees between a and the hypotenuse is less than 45 degrees. The values s = 6, t = 5 are also missing and would give triangle {11, 60, 61}.

Are they always primitive Pythagorean triples? 2 (H,S) Find all solutions in the positive integers to the equation x2 + y 2 = 1003. 3 (H,S) Find two primitive Pythagorean triples that represent triangles 20 Chapter 1 having different hypotenuses but equal area. 4 (H) Prove that for any integer n ≥ 3 there is a Pythagorean triangle with one of its legs having length n. For which integers n will there be a primitive Pythagorean triangle with n as the length of one of its legs? 5 (H,S) Prove that the radius of the inscribed circle of a Pythagorean triangle is always an integer.

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