Number Theory

Read e-book online An Introduction to Diophantine Equations: A Problem-Based PDF

By Titu Andreescu

This problem-solving publication is an advent to the research of Diophantine equations, a category of equations during which in simple terms integer recommendations are allowed. the fabric is prepared in components: half I introduces the reader to hassle-free equipment important in fixing Diophantine equations, reminiscent of the decomposition procedure, inequalities, the parametric technique, modular mathematics, mathematical induction, Fermat's approach to endless descent, and the tactic of quadratic fields; half II includes whole ideas to all routines partly I. The presentation good points a few classical Diophantine equations, together with linear, Pythagorean, and a few larger measure equations, in addition to exponential Diophantine equations. the various chosen workouts and difficulties are unique or are offered with unique solutions.

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

Show description

Read Online or Download An Introduction to Diophantine Equations: A Problem-Based Approach PDF

Similar number theory books

My Numbers, My Friends: Popular Lectures on Number Theory by Paulo Ribenboim PDF

This option of expository essays by means of Paulo Ribenboim may be of curiosity to mathematicians from all walks. Ribenboim, a hugely praised writer of a number of renowned titles, writes every one essay in a gentle and funny language with no secrets and techniques, making them completely available to every person with an curiosity in numbers.

New PDF release: Basiswissen Zahlentheorie: Eine Einführung in Zahlen und

Kenntnisse über den Aufbau des Zahlensystems und über elementare zahlentheoretische Prinzipien gehören zum unverzichtbaren Grundwissen in der Mathematik. Das vorliegende Buch spannt den Bogen vom Rechnen mit natürlichen Zahlen über Teilbarkeitseigenschaften und Kongruenzbetrachtungen bis hin zu zahlentheoretischen Funktionen und Anwendungen wie der Kryptographie und Zahlencodierung.

Get Model Theoretic Algebra With Particular Emphasis on Fields, PDF

This quantity highlights the hyperlinks among version idea and algebra. The paintings features a definitive account of algebraically compact modules, a subject of significant value for either module and version concept. utilizing concrete examples, specific emphasis is given to version theoretic strategies, resembling axiomizability.

Number theory through inquiry by Marshall, David C.; Odell, Edward; Starbird, Michael P PDF

Quantity conception via Inquiry; is an cutting edge textbook that leads scholars on a gently guided discovery of introductory quantity concept. The ebook has both major objectives. One aim is to aid scholars increase mathematical considering talents, rather, theorem-proving talents. the opposite aim is to aid scholars comprehend a number of the splendidly wealthy rules within the mathematical learn of numbers.

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

Example text

Remark. For n = 1, we get the equation √ 2x1 = x2 , and since √ 2 is irrational, there is no solution in this case. For n = 2, we have x22 x23 + x21 x23 = 3x21 x22 , or equivalently, a2 + b2 = 3c2 . We can assume that the numbers a, b, and c are all different from zero and that they are relatively prime, 44 Part I. Diophantine Equations meaning gcd(a, b, c) = 1. The square of an integer is congruent to 0 or 1 modulo 3, and hence both a and b are divisible by 3. Now, c is also divisible by 3 and we get a contradiction.

Note the solutions (0, k), k ∈ Z+ , and (1, 1). We will prove that there are no other solutions by using FMID on the prime 50 Part I. Diophantine Equations factors of x. Let p1 be a prime divisor of x and let q be the least positive integer such that p1 | 2q − 1. From Fermat’s Little Theorem we have p1 | 2p1 −1 − 1, and therefore q ≤ p1 − 1 < p1 . Let us prove now that q | x. If it didn’t, then x = kq + r, with 0 < r < q, and 2x − 1 = 2kq 2r − 1 = (2q )k · 2r − 1 = (2q − 1 + 1)k · 2r − 1 ≡ 2r − 1 (mod p1 ).

Prove that the following system of equations has infinitely many solutions in positive integers: ⎧ ⎨ ax − yz − c = 0, ⎩ bx − yt + d = 0. (Titu Andreescu) 28 Part I. Diophantine Equations 6. Find all triples (x, y, z) of integers such that xy(z + 1) = (x + 1)(y + 1)z. 7. Solve in integers the equation x2 + xy = y 2 + xz. 8. Prove that the equation x3 + y 3 + z 3 + w3 = 2008 has infinitely many solutions in integers. (Titu Andreescu) 9. Prove that there are infinitely many quadruples (x, y, z, w) of positive integers such that x4 + y 4 + z 4 = 2002w .

Download PDF sample

Rated 4.41 of 5 – based on 36 votes