100 Number Theory Problems (With Solutions)

Journal of Number Theory, 2008

We generalize Dirichlet's diophantine approximation theorem to approximating any real number α by a sum of two rational numbers a 1 q 1 + a 2 q 2 with denominators 1 ≤ q1, q2 ≤ N . This turns out to be related to the congruence equation problem xy ≡ c (mod q) with 1 ≤ x, y ≤ q 1/2+ǫ .

ISBN ().444.()()()71·2 250 Problems, in Elementary Number Theory .-WACLAW SIERPINSKI "250 Problems in Elementary Number Theory" presents problems and their solutions in five specific areas of this branch of mathematics: divisibility of numbers, relatively prime numbers, arithmetic progressions, prime and composite numbers, and Diophantic equations. There is, in addition, a section of miscellaneous problems. Included are problems on several levels of difficulty-some are relatively easy, others rather complex, and a number so abstruse that they originally were the subject of scientific research and their solutions are of comparatively recent date. All of the solutions are given thoroughly and in detail; they contain information on possible generalizations of the given problem and further indicate unsolved problems associated with the given problem and solution.