Download e-book for iPad: An Introduction to Diophantine Equations: A Problem-Based by Titu Andreescu

By Titu Andreescu

ISBN-10: 0817645489

ISBN-13: 9780817645489

ISBN-10: 0817645497

ISBN-13: 9780817645496

This problem-solving ebook is an advent to the research of Diophantine equations, a category of equations during which in simple terms integer ideas are allowed. the fabric is equipped in elements: half I introduces the reader to undemanding equipment useful in fixing Diophantine equations, reminiscent of the decomposition process, inequalities, the parametric procedure, modular mathematics, mathematical induction, Fermat's approach to endless descent, and the strategy of quadratic fields; half II comprises entire options to all workouts partially I. The presentation positive factors a few classical Diophantine equations, together with linear, Pythagorean, and a few larger measure equations, in addition to exponential Diophantine equations. some of the chosen routines and difficulties are unique or are provided with unique solutions.

An advent to Diophantine Equations: A Problem-Based Approach is meant for undergraduates, complicated highschool scholars and academics, mathematical contest individuals — together with Olympiad and Putnam rivals — in addition to readers attracted to crucial arithmetic. The paintings uniquely offers unconventional and non-routine examples, principles, and techniques.

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Extra info for An Introduction to Diophantine Equations: A Problem-Based Approach

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8. Prove that for any nonnegative integer k the equation x2 + y 2 − z 2 = k is solvable in positive integers x, y, z with x < y < z. 6 Fermat’s Method of Infinite Descent (FMID) 47 9. Prove that the equation x2 + (x + 1)2 = y 2 has infinitely many solutions in positive integers x, y. 10. Solve in distinct positive integers the equation x21 + x22 + · · · + x22002 = 1335(x1 + x2 + · · · + x2002 ). 6 Fermat’s Method of Infinite Descent (FMID) Pierre de Fermat (1601–1665) is famous for his contributions to mathematics even though he was considered only an amateur mathematician.

6) It is not known whether there are infinitely many positive integers n for which equation (1) admits solutions (x1 , x2 , . . , xn ), where x1 , x2 , . . , xn are all distinct odd positive integers. A simple parity argument shows that in this case n must be odd. There are several known examples of such integers n. For instance, if n = 9, we have 1 1 1 1 1 1 1 1 1 + + + + + + + + = 1; 3 5 7 9 11 15 33 45 385 42 Part I. Diophantine Equations if n = 11, 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + + = 1; 3 5 7 9 15 21 27 35 63 105 135 if n = 15, 1 1 1 1 1 1 1 1 1 + + + + + + + + 3 5 7 9 15 21 35 45 55 1 1 1 1 1 1 + + + + + = 1; + 77 165 231 385 495 693 and if n = 17, 1 1 1 1 1 1 1 1 1 + + + + + + + + 3 5 7 9 15 21 35 45 55 1 1 1 1 1 1 1 1 + + + + + + + = 1.

We have 1 1 1 = + ··· + , a3 (2a)3 (2a)3 where the right-hand side consists of eight summands, so if the equation (1) is solvable in positive integers, then so is the equation 1 1 1 + 3 + ··· + 3 = 1. 5 The Method of Mathematical Induction 45 412, 413, . . , 418. The key idea is to construct a solution in each of the above cases from smaller ones modulo 7. Observe that 27 33 9 36 4 + 3+ 3 3 2 3 6 32 4 + 3 23 4 18 243 + 3 33 9 18 16 144 + 3 + 3 33 4 12 16 36 144 4 + + + 23 43 63 123 = 1 and 27 ≡ 412 (mod 7), = 1 and 4 + 9 + 36 = 49 ≡ 413 = 1 and 4 + 32 = 36 ≡ 414 (mod 7), (mod 7), = 1 and 18 + 243 = 261 ≡ 415 (mod 7), = 1 and 18 + 16 + 144 = 178 ≡ 416 = 1 and 4+16+36+144 = 200 ≡ 417 (mod 7), (mod 7).

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An Introduction to Diophantine Equations: A Problem-Based Approach by Titu Andreescu


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