Lectures on Number TheoryThis volume is a translation of Dirichlet's Vorlesungen uber Zahlentheorie which includes nine supplements by Dedekind and an introduction by John Stillwell, who translated the volume. Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions. The book is suitable as a textbook, yet it also offers a fascinating historical perspective that links Gauss with modern number theory. |
Contents
| xi | |
| 1 | |
On the congruence of numbers | 21 |
On quadratic residues | 53 |
On quadratic forms | 91 |
Determination of the class number of binary quadratic forms | 149 |
Some theorems from Gausss theory of circle division | 199 |
On the limiting value of an infinite series | 211 |
A geometric theorem | 215 |
Genera of quadratic forms | 217 |
Power residues for composite moduli | 229 |
Primes in arithmetic progressions | 237 |
Some theorems from the theory of circle division | 249 |
On the Pell equation | 257 |
Convergence and continuity of some infinite series | 261 |
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Lectures on Number Theory Peter Gustav Lejeune Dirichlet,Peter Dirichlet,Richard Dedekind Limited preview - 1999 |
Common terms and phrases
absolute value belong class number complete residue system congruence continued fraction Conversely corresponding Crelle's Journal cy² decomposition denote Dirichlet 2nd edition Du² equation equivalent example exponent footnote from Dirichlet formula Gauss D.A. art generalised given numbers greatest common divisor hence incongruent numbers incongruent roots infinite series latter least common multiple left hand side Legendre symbol limiting value modulus multiple negative determinant neighbouring form nonzero number of terms number theory numbers n numbers relatively prime obtain obviously odd number odd prime pairs Pell equation positive determinant positive integer positive value prime factors prime to 2D primes dividing primitive root proof proved quadratic forms quadratic nonresidue quadratic residue reciprocity theorem reduced forms representations result right hand side satisfy solution square substitution tends to zero Translator's note Wilson's theorem σ σ
Popular passages
Page xiii - Legendre's statement of the theorem — which is to say that/? is a square mod q if and only if q is a square mod p...
Page 62 - But neither of them was able to prove the theorem, and Waring confessed that the demonstration seemed more difficult because no notation can be devised to express a prime number. But in our opinion truths of this kind should be drawn from notions rather than from notations.
Page 3 - The logarithms of negative factors in any group are found without regard to the negative signs, and the result is positive or negative according as the number of negative factors is even or odd.
Page 20 - It is now clear that the whole structure rests on a single foundation, namely the algorithm for finding the greatest common divisor of two numbers. All the subsequent theorems, even when they depend on the later concepts of relative and absolute prime numbers, are still only simple consequences of the result of this initial investigation...
Page 63 - The number -1 is a quadratic residue of all primes of the form 4n + 1 and a quadratic non-residue of all primes of the form 4n + 3.
Page 257 - Proposition 10 that, if d is a positive integer which is not a perfect square, then the equation (8) always has a solution in positive integers and all such solutions are given by * =ptt-i. y = <ikh-i (* = 1-2-- ) if h is even' x=P2kh-l> y = l2kh-\ (* = 1.2,...
Page xvi - It is greatly to be lamented that this virtue of the real numbers [that is, of the ordinary integers] to be decomposable into prime factors, always the same ones for a given number, does not also belong to the complex numbers...
Page xv - ... he was completing the section on composition of forms in the Disquisitiones Arithmeticae. but that he was never able to put it on a firm foundation; he says in particular, in a note to his article on the decomposition of polynomials into linear factors, that: 'If I wanted to proceed with the use of imaginaries in the way that earlier mathematicians have done, then one of my earlier researches which is very difficult could have been done in a very simple way.
Page 63 - ... especially in VLSI implementation. QRNS structure has been employed in Fermat Number Transform by Nussbaumer [14], but it has not been proposed for general complex DSP algorithms except recently by Leung [5]. The algebraic foundation for QRNS is based on the following theorem [15].