to successfully present the required number of lectures,  
to actively participate in the discussions (in particular it implies ones presence in the meetings!),  
to reasonably treat at least 70% of the proposed exercises and to present some of the solutions to the audience. 
Lectures:  Exercises:  
Lecture 1, Tuesday, February 19, 2013, Speaker: Oleksandr Iena Topic: Greatest common divisor for integer numbers and linear Diophantine eqiations. 
Exercises from Lecture 1 psfile , pdffile  
Lecture 2, Tuesday, February 26, 2013, Speaker: Philippe Dieschburg Topic: Primality tests. 
Exercises from Lecture 2 psfile , pdffile  
Lecture 3, Tuesday, March 5, 2013, Speaker: Paul Schmitz Topic: Continued fractions. 
Exercises from Lecture 3 psfile , pdffile  
Lecture 4, Tuesday, March 12, 2013, Speaker: Selma Skelic Topic: Sums of squares. 
Exercises from Lecture 4 psfile , pdffile  
Lecture 5, Tuesday, March 19, 2013, Speaker: Marine Rabbottini Topic: RSA algorithm. 
Exercises from Lecture 5 psfile , pdffile  
Lecture 6, Tuesday, March 26, 2013, Speaker: Morgane Mencagli Topic: ElGamal Algorithm. 
Exercises from Lecture 6 psfile , pdffile  
Lecture 7, Tuesday, April 9, 2013, Speaker: Sven Kill Topic: padic numbers. 
Exercises from Lecture 7
psfile
,
pdffile
Solutions of the second part of the last exercise: pdffile 

Lecture 8, Tuesday, April 16, 2013, Speakers: Paul Schmitz & Selma Skelic Topic: Quadratic residues. 
Exercises from Lecture 8 psfile , pdffile  
Lecture 9, Tuesday, April 23, 2013, Speakers: Morgane Mencagli & Marine Rabbottini Topic: Fibonacci numbers. 
Exercises from Lecture 9 psfile , pdffile  
Lecture 10, Tuesday, April 30, 2013, Speaker: Sven Kill Topic: TonelliShanks algorithm. 
Exercises from Lecture 10 psfile , pdffile  
Lecture 11, Tuesday, May 7, 2013 Exercise session 
Exercises from Lecture 11 psfile , pdffile  
Lecture 12, Tuesday, May 14, 2013 Exercise session, Babystep gianttep algorithm. 
Exercises from Lecture 12 psfile , pdffile  
Lecture 13, Tuesday, May 21, 2013 Exercise session. 
Exercises from Lecture 13 psfile , pdffile 
A Friendly Introduction to Number Theory.


A nice note about sums of two squares.

Main books:  
Silverman, Joseph H.  A friendly introduction to number theory, Upper Saddle River, N.J. : Pearson, 2006, 264 p.  
MüllerStach, Stefan; Piontkowski, Jens.  Elementare und algebraische Zahlentheorie. Ein moderner Zugang zu klassischen Themen, Wiesbaden: Vieweg. 2011, ix, 240 p.  
Baldoni, Maria Welleda; Ciliberto, Ciro; Piacentini Cattaneo, Giulia Maria.  Elementary number theory, cryptography and codes, Berlin : Springer, 2009.  522 p.  
Cohen, Henri.  A course in computational algebraic number theory, Berlin: Springer, 1995, 534 p.  
Cox, David A.  Primes of the form $x^2+ny^2$. Fermat, class field theory, and complex multiplication, WileyInterscience Series in Pure and Applied Mathematics. New York, NY: Wiley, 1997, xi, 351 p.  
Additional books:  
Neukirch, Jürgen.  Algebraic number theory; transl. from the German by Norbert Schappacher, Berlin: Springer, 1999, 571 p.  
Samuel, Pierre.  Algebraic theory of numbers. (Translated from the French by Allan J. Silberger), Boston, Mass.: Houghton Mifflin Co., 1970, 109 p.  
Stillwell, John.  Elements of number theory, New York, NY: Springer. 2003, xii, 254~p.  
Pollard, Harry.  The theory of algebraic numbers, Carus Mathematical Monographs, No.9. New York: John Wiley & Sons. 1950, XII, 143 p. 