This book is also an introduction to the topic of 'elementary methods in analytic number theory'. The theorems in this book are simple statements about integers, but the standard proofs require contour integration. The integer of the real number x, denoted by [x], is the largest integer that is less than or equal to x.
A First Course in Number Theory
Division Algorithm
We can see the necessity of the condition that the nonempty set is bounded below by considering the example of the setZof all integers, positive, negative, and zero. Another form of the principle of mathematical induction says that ifS(k0) is true and if the truth of S(k0), S(k0+ 1),.
Greatest Common Divisors
Prove that the set of all rational numbers of the form a/2k, where a∈Zandk∈N0, is an additive subgroup of the group Q. Determine the addition of elements Gby. Prove that H is a subgroup if and only if x−y∈H holds for allx, y∈H. Prove that if G is a group written multiplicatively and g∈G, then gn ∈G for all n∈Z. If G is an additive group, then ng∈G holds for all n∈Z.).
The Euclidean Algorithm and Continued Fractions
We can use the Euclidean algorithm to write a rational number as a finite simple continued fraction with integral part quotients. Note that the partial quotients in the Euclidean algorithm are the partial quotients in the continued fraction.
The Fundamental Theorem of Arithmetic
Prove that every positive integer can be written uniquely as the product of a square and a square-free integer. Prove that every powerful number can be written as the product of a square and a cube.
Euclid’s Theorem and the Sieve of Eratosthenes
We can calculate all the prime numbers to poison using a beautiful and efficient method called the sieve of Eratosthenes. Change the proof of Theorem 1.14 to prove that there are infinitely many prime numbers whose remainder is 3 when divided by 4.
A Linear Diophantine Equation
The following theorem solves the Frobenius problem in the case k= 2. Theorem 1.17 Let a1 anda2 be relatively prime positive integers. We saw in the proof of Theorem 1.15 that for every integer b there exists integerx1andx2 such that.
Notes
Guy's Unsolved Problems in Number Theory[45] is a nice survey of unusual problems and results in elementary number theory. An Internet site devoted to Mersenne primes and related problems in number theory is www.mersenne.org.
The Ring of Congruence Classes
From these identities we see that the set of congruence classes modulo m is an Abelian group under addition. The congruence class 1 +mZ is a multiplication identity, since (1 +mZ)(a+mZ) =a+mZ. Finally, multiplication of congruence classes is distributive with respect to addition in the sense that .
Linear Congruences
If a is an integer relatively prime, then there exists an integer such that a≡a (modd)anda is relatively prime. In particular, if a is relatively prime, then every integer in the congruence class a+mZ is relatively prime to m. A congruence class of modulo is called relatively prime if some (and consequently every) integer in the class is relatively prime.
The Euler Phi Function
Since there are modulomn exactly different congruence classes, the congruence (2.4) has a unique solution for each integer. By Theorem 2.6, each congruence class modulo mn can be written uniquely in the form ma+nb+mnZ, where a and b are integers such that 0≤a ≤n−1 and 0≤b ≤m−1. Since there are ϕ(n) integerssa∈[0, n−1] that are relatively prime for n, and ϕ(m) integers b ∈ [0, m−1] that are relatively prime for m, it follows thatϕ( mn) =ϕ (m)ϕ(n), and so the Euler phi function is multiplicative.
Chinese Remainder Theorem
If y is another solution of the system of k congruences, then x−y is divisible by mi for alli= 1,. There is an important application of the Chinese remainder theorem to the problem of solving Diophantine equations of the form. The Chinese remainder theorem allows us to reduce the question of the solvability of this congruence moduli to the special case of primary power modulipr.
Euler’s Theorem and Fermat’s Theorem
This follows immediately from Theorem 2.15, since the order of a is the order of the cyclic subgroup that generates.2. Let (a, m) = 1 and let be the order of a+mZ in G, that is, the order of the cyclic subgroup generated by a+mZ. Prove that the decimal expansion of the fraction 1/mis is periodic with period equal to the order of 10 modulom.
Pseudoprimes and Carmichael Numbers
This test can prove that an integer is composite, but it cannot prove that an integer is prime. A Carmichael number is a positive integer n such that n is composite butbn−1≡1 (mod n) for every integer relative prime. Prove that there exists a nonnegative integer such that n+u2= (u+ 1)2. Prove that it is composed if and only if there exist nonnegative integers such that v > u+ 1 enn+u2=v2.
Public Key Cryptography
It is important to note that we reveal neither ϕ(m) nor the prime factors p and q of m. If the prime numbers p and q are large (such as several thousand digits each), then it is impossible with the latest computer hardware and our current knowledge about factoring large numbers to find the prime factors of a reasonable amount of time, say, a million years. The following result indeed shows that knowing ϕ(m) is equivalent to knowing the prime factors of m.
Notes
Polynomials and Primitive Roots
By Theorem 3.2, the polynomial (x) =xd−1∈F[x] has at most zeros, and thus every zero of f(x) belongs to the cyclic subgroup a. Prove that if there is a primitive root, then there exist exactly ϕ(ϕ(m)) pairwise incompatible primitive roots modulo. Prove that ifI={0} is an ideal of the polynomial ringF[x], where F is a field, then there exists a unique monic polynomial (x)∈I such that I consists of all multiples ofd(x), i.e. . .
Primitive Roots to Composite Moduli
If g is a primitive root modulopk and g1∈ {g, g+pk} is odd, then ng1 is a primitive root modulo2pk. For example, 2 is a primitive root modulo 3. Finally, we consider the modulo primitive roots of 2. Theorem 3.8 There exists a primitive root modulo = 2k if and only if m= 2 or4. Find an integer g that is a primitive root modulo 5k for allk ≥ 1. Prove that it divides the binomial coefficientp.
Power Residues
It is an important unsolved problem in number theory to understand the distribution of fractional parts of powers of 3/2 in the interval [0,1). We can check that 2 is a primitive root modulo 19, and is a cube residue modulo 19 if and only if 3 divides ind2(a). Prove that f is an isomorphism of the multiplicative group (Z/23Z)×, that is, prove that f is a homomorphism that is one-to-one and onto.
Quadratic Residues
Again the statement follows immediately from the observation that both sides of this congruence are ± 1.2. A prime number of the form Mp is called a Mersenne prime (see exercise 5 in section 1.5). a) Prove that 2 has order pmoduloq, and divides sop q−1. Prove that f(x, y) has a non-trivial solution modulopif and only if d≡0 (modp) or dis is a quadratic residue modulop.
Quadratic Reciprocity Law
The quadratic reciprocal law provides an effective method for calculating the value of the Legendre symbol. Quadratic reciprocity also enables us to determine all primes for which a given integer a is a quadratic residue. Using the notation in the proof of the law of quadratic reciprocity (Theorem 3.17), we have m+n+M+N.
Quadratic Residues to Composite Moduli
Hence there exists an integer1 such that f(x1+y1p)≡0 (modp2) if and only if it is a linear congruence. According to Hensl's lemma, the polynomial congruence f(x) ≡ 0 (mod pk) is solvable for every k ≥1, so a is a quadratic residue modulopk for every k≥1,2. Prove that if (x) is a polynomial with integer coefficients, then D(k)(f)(x) = 0 if and only if the degree of off(x) is at most k−1.
Notes
The Structure of Finite Abelian Groups
Let g1 ∈G be an element of the highest order pr1 and let G1=g1 be the cyclic subgroup generated by g1. On the other hand, g+G1 has order pr in the quotient group G/G1, so the order of g is at least pr. If G is not cyclic, let g1 ∈ G be an element of highest order pr1 and let G1 be the cyclic subgroup generated by g1.
Characters of Finite Abelian Groups
By Lemma 4.2, the dual of a finite cyclic group of order is also a finite cyclic group of order n. For example, if C4 is a cyclic group of order 4 with generator g0, then the signs of C4 are functions. Let p be a prime number and let G= (Z/pZ)× be the multiplicative group of units in the field Z/pZ. a) Prove that χa is a character, i.e. χa ∈G.
Elementary Fourier Analysis
Let G denote the dual dual of G, i.e. the group of characters of the dual group G. Applying the Cauchy-Schwarz inequality (4.5) withf1=f(χ) and withf2 the characteristic function of the set supp. Prove that a product of Fourier transforms is the convolution of the product in the sense that.
Poisson Summation
This is another example showing that the lower bound in the uncertainty principle (Theorem 4.10) is the best possible one.
Trace Formulae on Finite Abelian Groups
It follows that we can define the trace of a linear operator T on a vector space V as the trace of the matrix of T with respect to some basis for V, and that this definition does not depend on the choice of the basis. By Theorem 4.12, since Gi is a basis for L2(G), the trace of Ch is the sum of the eigenvalues. There exists an orthonormal basis for the vector space L2(G/H) consisting of the functions δx, where.
Gauss Sums and Quadratic Reciprocity
If it is a multiplicative character of Z/mZ, then we extend to a function onZ/mZ by defining χ(a+mZ) = 0 if (a, m)= 1. For each integer and multiplicative characterχ, we define the sum of Gaussian τ(χ, a) as the Fourier transform of χ evaluated with the additional character ψ−a, so. From Exercise 10 in Section 4.3, a product of Fourier transforms is the Fourier transform of the convolution, and so.
The Sign of the Gauss Sum
Since the determinant and the trace of a linear operator on a finite-dimensional vector space are independent of the choice of basis for the vector space, it follows that the trace of the Fourier transform F on the group Z/nZ is the complex conjugate of Gauss sumτ( n). We need to calculate the matrix of the Fourier transform F with respect to this basis. whereχ0 is the primary multiplicative sign modulo p. 4.24). Prove that the Legendre symbol is an eigenvector of the Fourier transform with eigenvalue (−1)(p−1)/2τ(p).
Notes
Ideals and Radicals
The prime ideals in the ring Z are thus the ideals of the form pZ, in which a prime number orp= 0.2. In exercise 9, the radical of the ring is the point of intersection between the primary ideals in the ring. Thus the prime ideals of the ringZ/mZ are the ideals of form+mZ, where pi is a prime divisor of m.
The ring C[t] of polynomials with coefficients in the field C of complex numbers is a principal ring
- Derivations
- Mason’s Theorem
- The abc Conjecture
Prove that the radical of the ring R is the intersection of all prime ideals of R, that is. We also say that S−1R is constructed by RatS localization. b) Prove that if R is an integral domain and 0∈S, then S−1R is an integral domain. Prove that S is a multiplicative subset of Z, and describe the ring of fractions S−1Z. Prove that the principal ideal generated by the maximal ideal in S−1Z. Let F[t] be a polynomial ring with coefficients in the field F. Let F[t] be a multiplicative subset consisting of powers oft.
