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

Number Theory

By Titu Andreescu

ISBN-10: 0817645489

ISBN-13: 9780817645489

This problem-solving ebook is an advent to the learn of Diophantine equations, a category of equations during which in simple terms integer ideas are allowed. the fabric is prepared in components: half I introduces the reader to user-friendly tools invaluable in fixing Diophantine equations, resembling the decomposition approach, inequalities, the parametric procedure, modular mathematics, mathematical induction, Fermat's approach to countless descent, and the strategy of quadratic fields; half II comprises whole ideas to all routines partly I. The presentation positive aspects a few classical Diophantine equations, together with linear, Pythagorean, and a few greater measure equations, in addition to exponential Diophantine equations. a few of the chosen workouts and difficulties are unique or are offered with unique solutions.

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

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3 The Parametric Method 27 Exercises and Problems 1. Prove that the equation x2 = y 3 + z 5 has infinitely many solutions in positive integers. 2. Show that the equation x2 + y 2 = z 5 + z has infinitely many solutions in relatively prime integers. (United Kingdom Mathematical Olympiad) 3. Prove that for each integer n ≥ 2 the equation xn + y n = z n+1 has infinitely many solutions in positive integers. 4. Let n be an integer greater than 2. Prove that the equation xn + y n + z n + un = v n−1 has infinitely many solutions (x, y, z, u, v) in positive integers.

Solve in positive integers the equation xy + y = y x + x. 14. Let a and b be positive integers such that ab+1 divides a2 +b2 . Prove that a2 +b2 ab+1 is the square of an integer. (29th IMO) 15. Find all integers n for which the equation (x + y + z)2 = nxyz is solvable in positive integers. (American Mathematical Monthly, reformulation) 20 Part I. 3 The Parametric Method In many situations the integral solutions to a Diophantine equation f (x1 , x2 , . . , xn ) = 0 can be represented in a parametric form as follows: x1 = g1 (k1 , .

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. + 77 165 275 385 495 825 1925 2475 Example 4. Prove that equation 1 1 n+1 1 + 2 + ··· + 2 = 2 2 xn x1 x2 xn+1 is solvable in positive integers if and only if n ≥ 3.

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