An Introduction to the Theory of Numbers
"This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text."--Publisher's description.
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addition chain admissible composition algebraic arithmetic functions arithmetic progression Brauer Chowla class will contain combinatorial number theory combinatorial principle congruences cos1 deﬁned deﬁnition diﬀerent diﬃcult Diophantine Equation Dirichlet series distinct a’s elementary elements ep(n Erdos estimate Euler example exist extended Riemann hypothesis Fermat prime Fermat’s ﬁeld ﬁnd ﬁnding ﬁnite number ﬁrst digit fundamental theorem Hence implies infinitely inﬁnitely many primes integer coefficients integral coeﬃcients irrational irreducible polynomial least nonresidue Lemma Let a1 log2 loglogn logn logp modp Moser nonaveraging number of compositions number of distinct number of integers number of partitions number of solutions obtain pn(c polynomial with integer positive integers prime factors Prime Number Theorem problem Prove Publisher’s note quadratic residues relatively prime representation result Schur’s theorem sequence Show Sn(m solvable square suﬃciently large sum of distinct Suppose theorem of Minkowski transcendental numbers unsolved visible lattice points Waerden’s yields
Page 4 - The number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts. For
Page 50 - classes in any manner whatsoever, at least one of the classes will contain an arithmetic progression of
Page 4 - 0(n) the number of partitions of n into an odd number of distinct parts. We
Page 4 - parts is equal to the number of partitions of n into parts
Page 55 - 2 it follows that m — 1 must have at least one prime divisor, so by (4) n is even. We now multiply together all congruences of the type (5) , that is one for each prime dividing m — 1. Since m — 1 is squarefree, the resulting modulus is
Page 55 - The only values of m ¿ 1000 which satisfy (8) are 3, 7, 43. We proceed to develop three more congruences, similar to (8) , which when combined with (8) lead to the main result. Equation (1) can be written in the form
Page 4 - of the graph. From C, the extreme north-east node, we draw the longest south-westerly line possible in the graph; this
Page 54 - and n. This is essentially our method, but the moduli are so chosen that we are able to combine the resulting congruences so as to obtain extremely large bounds for