• Aucun résultat trouvé

Phnom Penh, RUPP Final Exam: April 6, 2012

N/A
N/A
Info
Télécharger
Protected

Academic year: 2022

Partager "Phnom Penh, RUPP Final Exam: April 6, 2012"

Copied!
1
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)

Phnom Penh, RUPP Final Exam: April 6, 2012

RUPP Master in Mathematics Program: Number Theory Michel Waldschmidt

1. Prove by induction that 4

2n+1

+ 3

n+2

is a multiple of 13 for n ≥ 0.

2. If m > 1 and a is prime to m, show that the remainders obtained by dividing a, 2a, . . . , (m − 1)a by m are the numbers 1, 2, . . . , m − 1 in some order.

3. If m is any odd integer, prove that

1

m

+ 2

m

+ · · · + (m − 1)

m

≡ 0 (mod m).

4. Let G be a finite multiplicative group with p elements and p is prime.

Show that G is cyclic and that any element 6= 1 is a generator.

5. Show that a finite ring without zero divisor is a field.

6.

a) Find all solutions (x, y) in positive integers of the equation x

2

− 7y

2

= −1.

b) Find two solutions (x

1

, y

1

) and (x

2

, y

2

) in positive integers with 1 < x

1

<

x

2

of the equation x

2

− 7y

2

= 1.

Hint:

the following computations can be used:

√

7 = 2.6457· · ·= 2 + 1 x1

, x1= 1.5485· · ·= 1 + 1 x2

, x2= 1.8228· · ·= 1 + 1

x3

, x3 = 1.2152· · ·= 1 + 1 x4

, x4 = 4.6457· · ·= 2 +√

7.

http://www.math.jussieu.fr/∼miw/enseignement.html

Références

Télécharger maintenant ( PDF - 1 Page - 117.80 KB )

Documents relatifs

Exercise 1. Let b ≥ 2 be an integer. Show that a real number x is rational if and only if the sequence (d

Write a degree 2 polynomial with integer coefficients having a root at the real number whose continued fraction expansion is. [0; a,

Exercise 1. [(8.31) [1]] Show that if ρ : R → G is a homomorphism from ( R , +) to (G, ·), then it is of the form ϕ

University of Zurich (UZH) Fall 2012 MAT532 – Representation theory..

Seminar in Number Theory, HKUST, February 2006 Michel Waldschmidt, Professor, University Paris VI

Seventh lecture: Monday, February 27, 2006, 3-4 pm Room 2404 Lift 17 &amp; 18 The role of complex conjugation in transcendental number theory. One of Ramachandra's contributions

How to prove that a number is irrational?

It is expected that the result is not true with = 0 as soon as the degree of α is ≥ 3, which means that it is expected no real algebraic number of degree at least 3 is

2. Let p and q be distinct odd primes. Show that p q−1 + q p−1 ≡ 1 (mod pq).

RUPP Masters in Mathematics Program: Number Theory.. Problem Set

Report on my visit to Phnom Penh, April 20 - May 8, 2015 RUPP - Royal University of Phnom Penh

From April 20 to May 8, 2015, I taught a 45 hours course Algebra and Geometry to the students of year 1 of the Master of Science in Mathematics of the Royal University of Phnom

February 3, 2012 Centre of Excellence in Mathematics (CEM) Department of Mathematics, Mahidol University Transcendental Number Theory: recent results and open problems. Michel Waldschmidt

This conjecture, which is similar in spirit to the Hodge conjecture, is one of the central conjectures about algebraic independence and transcendental numbers, and is related to many

Transcendental Number Theory: recent results and open problems. Michel Waldschmidt

A period is a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients, over domains in R n given