An Introduction to Number Theory (Graduate Texts in Mathematics, Vol. 232)

An Introduction to Number Theory (Graduate Texts in Mathematics, Vol. 232)

Language: English

Pages: 297

ISBN: 1852339179

Format: PDF / Kindle (mobi) / ePub


Includes up-to-date material on recent developments and topics of significant interest, such as elliptic functions and the new primality test

Selects material from both the algebraic and analytic disciplines, presenting several different proofs of a single result to illustrate the differing viewpoints and give good insight

Trigonometric Delights (Princeton Science Library)

Algebraic Geometry and Commutative Algebra (Universitext)

College Trigonometry (6th Edition)

An Introduction to Manifolds (2nd Edition) (Universitext)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fermat’s Little Theorem, 210 ≡ 1 modulo 11 so 2340 ≡ 134 ≡ 1 modulo 11. Also 25 = 32 ≡ 1 modulo 31, so 2340 = (25 )68 ≡ 168 = 1 (mod 31). Thus 2340 − 1 is divisible by the coprime numbers 11 and 31, and hence by their product 341, so 2340 ≡ 1 modulo 341. However, Fermat’s Little Theorem says more than Equation (1.26): It gives the congruence ap−1 ≡ 1 (mod p) for any base a, not just a = 2. Taking a = 3 in Example 1.20, we quickly find 3340 ≡ 56 (mod 341), which contradicts Fermat’s Little Theorem

number DE measured by A, B, and C. Add the unit DF to DE. Then EF is either prime or not. First, let it be prime. Then the prime numbers A, B, C, and EF have been found, which are more than A, B, and C. Next, let EF not be prime. Therefore, it is measured by some prime number. Let it be measured by the prime number G. I say that G is not the same as any of the numbers A, B, and C. 40 1 A Brief History of Prime If possible, let it be so. Now A, B, and C measure DE, and therefore G also

the remarkable properties of elliptic curves. Exercise 2.14. Prove that the polynomial x3 + ax + b has no repeated zero if and only if 4a3 + 27b2 = 0. The genius of people such as Siegel is that they are willing to take an imaginative step up from particular cases, and are in addition able to supply the guile needed to complete the proof. In fact, he gave two different proofs of Theorem 2.13. In his second proof Siegel showed that there are only finitely many solutions (x, y) with x and y lying

on deep results concerning the arithmetic of elliptic curves: If ap + bp + cp = 0 for a prime p and integers a, b, c, then the elliptic curve with equation y 2 = x(x − ap )(x + bp ) turns out to have properties that Wiles was able to show were impossible. We will be studying the arithmetic of elliptic curves in Chapters 5 and 6. 2.5 Fermat, Catalan, and Euler 57 Exercise 2.19. *Prove that Equation (2.12) has no nontrivial solutions with n equal to 3, 4, or 5. Exercise 2.20. *Prove that

Stevens, Alan and Honor Ward, and others for pointing out errors and suggesting improvements. Errors and solecisms that remain are entirely the authors’ responsibility. February 14, 2005 Norwich, UK Graham Everest Thomas Ward Notation and terminology “Arithmetic” is used both as a noun and an adjective. The particular notation used is collected at the start of the index. The symbols N, P, Z, Q, R, C denote the natural numbers {1, 2, 3, . . . }, prime numbers {2, 3, 5, 7, . . . }, integers,

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