Abstract: Tore Herlestam, in his note "Critical Remarks on Some Public-Key Cryptosystems", [5] suggests a method for attacking the RSA public-key cryptosystem. In this note we show that Herlestam's proposed attack is highly impractical, and that his analysis is erroneous. The RSA cryptosystem [1] encodes a message M using the key (e,n) via the equation: (1) C =E ~ ( M ) M e (modn) . Here the original message M and the ciphertext C are considered as integers in the range 0 to n 1. The integer n is the product of two large prime numbers p and q. The integer e is relatively prime to ( p 1 ) ( q 1 ) . To decrypt a received ciphertext C the recipient computes (2) M Dd,(M) C d (modn) where d is chosen to satisfy the equation de --1 ( m o d l c m ( p l , q 1 ) ) . The attack proposed by Herlestam runs as follows: Let P(x) be a polynomial in x such that (3) P ( x ) = x Q ( x ) .

Abstract: Volume I.- About This Book.- 1. Necessary Results from Measure Theory.- Steinhaus' Lemma.- Cauchy's Functional Equation.- Slowly Oscillating Functions.- Halasz' Lemma.- Fourier Analysis on the Line: Plancherel's Theory.- The Theory of Probability.- Weak Convergence.- Levy's Metric.- Characteristic Functions.- Random Variables.- Concentration Functions.- Infinite Convolutions.- Kolmogorov's Inequality.- Levy's Continuity Criterion.- Purity of Type.- Wiener's Continuity Criterion.- Infinitely Divisible Laws.- Convergence of Infinitely Divisible Laws.- Limit Theorems for Sums of Independent Infinitesimal Random Variables.- Analytic Characteristic Functions.- The Method of Moments.- Mellin - Stieltjes Transforms.- Distribution Functions (mod 1).- Quantitative Fourier Inversion.- Berry-Esseen Theorem.- Concluding Remarks.- 2. Arithmetical Results, Dirichlet Series.- Selberg's Sieve Method a Fundamental Lemma.- Upper Bound.- Lower Bound.- Distribution of Prime Numbers.- Dirichlet Series.- Euler Products.- Riemann Zeta Function.- Wiener-Ikehara Tauberian Theorem.- Hardy-Littlewood Tauberian Theorem.- Quadratic Class Number, Dirichlet's Identity.- Concluding Remarks.- 3. Finite Probability Spaces.- The Model of Kubilius.- Large Deviation Inequality.- A General Model.- Multiplicative Functions.- Concluding Remarks.- 4. The Turan-Kubilius Inequality and Its Dual.- A Principle of Duality.- The Least Pair of Quadratic Non-Residues (mod p).- Further Inequalities.- More on the Duality Principle.- The Large Sieve.- An Application of the Large Sieve.- Concluding Remarks.- 5. The Erdos-Wintner Theorem.- The Erdos-Wintner Theorem.- Examples ?(n),?(n).- Limiting Distributions with Finite Mean and Variance.- The Function ?(n).- Modulus of Continuity, an Example of an Erdos Proof.- Commentary on Erdos' Proof.- Concluding Remarks.- Alternative Proof of the Continuity of the Limit Law.- 6. Theorems of Delange, Wirsing, and Halasz.- Statement of the Main Theorems.- Application of Parseval's Formula.- Montgomery's Lemma.- Product Representation of Dirichlet Series (Lemma 6.6).- Quantitative form of Halasz' Theorem for Mean-Value Zero.- Concluding Remarks.- 7. Translates of Additive and Multiplicative Functions.- Translates of Additive Functions.- Finitely Distributed Additive Functions.- The Surrealistic Continuity Theorem (Theorem 7.3).- Additive Functions with Finite First and Second Means.- Distribution of Multiplicative Functions.- Criterion for Essential Vanishing.- Modified-weak Convergence.- Main Theorems for Multiplicative Functions.- Examples.- Concluding Remarks.- 8. Distribution of Additive Functions (mod 1).- Existence of Limiting Distributions.- Erdos' Conjecture.- The Nature of the Limit Law.- The Application of Schnirelmann Density.- Falsity of Erdos' Conjecture.- Translation of Additive Functions (mod 1), Existence of Limiting Distribution.- Concluding Remarks.- 9. Mean Values of Multiplicative Functions, Halasz' Method.- Halasz' Main Theorem (Theorem (9.1)).- Halasz' Lemma (Lemma (9.4)).- Connections with the Large Sieve.- Halasz's Second Lemma (Lemma (9.5)).- Quantitative Form of Perron's Theorem (Lemma (9.6)).- Proof of Theorem (9.1).- Remarks.- 10. Multiplicative Functions with First and Second Means.- Statement of the Main Result (Theorem 10.1).- Outline of the Argument.- Application of the Dual of the Turan-Kubilius Inequality.- Study of Dirichlet Series.- Removal of the Condition p > p0.- Application of a Method of Halasz.- Application of the Hardy-Little wood Tauberian Theorem.- Application of a Theorem of Halasz.- Conclusion of Proof.- Concluding Remarks.- References (Roman).- References (Cyrillic).- Author Index xxm.

Abstract: A positive integern is called a pseudoprime ifn|2 n −2 andn is composite. W. Sierpinski put forward the following problem: Do there infinitely many arithmetical progressions formed of four pseudoprimes? In this paper it is proved that answer to this problem is in the affirmative.

Abstract: By making use of Hasse's sum theorem, a simple proof of the following theorem on Schur indices of p-groups is given. THEOREM (ROQUETTE [3] AND SOLOMON [4]). Let p be a prime number, G a p-group, and X an irreducible complex character of G. Let mQ(X) denote the Schur index of X over the rational field Q. Then, mQ(X) = 1 for p # 2, and mQ(X) = 1 or 2 for p = 2. PROOF. Let A be the simple component of the group algebra Q[G], which corresponds to X. The center of A is Q(X), the extension field of Q generated by the elements {x(g); g C G}. Put k = Q(X). Let q be a rational prime (possibly the infinite prime oo) and q a prime of k, lying above q. Let inv,(X) denote the Hasse invariant of A at q. It is well known that if q :#p, oo, then invq(X)= 0 (mod 1), i.e., the Schur index mQ,() = 1, Qq being the q-adic numbers. (A result which may be established by means of modular representation theory for the prime q I G 1.) Let IGI = pfn and D a primitivepnth root of unity. Then k = Q(X) c Q('). Hence there is only one prime p of k lying above p (cf. Theorem 1 of [2, p. 73]). Let P0.1, ... . PO s be the infinite primes of k. Hasse's sum theorem (Satz 9, p. 119 of [1]) now yields that invp(X) + E invp (x) 0 (mod 1). i=1 Since invp (X) 0 or 2 it follows that Es. linvp (X) 0 or 2 (mod 1), and consequently invj(X)= 0 or 2 (mod 1). This implies that mQ(X) = 1 or 2. Since mQ(X)Ipn, then mQ(X) = 1 forp =# 2.

Abstract: The problem of finding prime numbers has been much favoured by expositors of modern programming methodology. In this paper the problem is tackled once more, but from a broader view of the programming process than has previously been taken. The initial steps toward a solution parallel those in the well-known presentations of Dijkstra and Wirth. However a general program transformation is shown to capture the clever "inventive step" in Dijkstra's solution, and another is employed to lead naturally to an alternative program. Complexity analyses of the two programs show that the latter is significantly more efficient.

Abstract: Matatyahu Rubin pointed out that the proof of Lemma 6.1 [2] works only for rings of prime or zero characteristic. This invalidates the characterization of semiprime rings with the descending chain condition on right or left ideals which admit elimination of quantifiers given in [2] and cited in the abstract [1]. Although the correct characterization is easy to derive, it is complex to state. Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF ( p n ) ⊕ GF ( p k ) such that either n = k or g.c.d.( n, k ) = 1 and p is a prime. Let ′ be the class of algebraically closed fields. Let P denote the set of all prime numbers together with zero. Let be the set of all ordered pairs ( f, Q ) where Q is a finite subset of P and f : Q → ⋃ ⋃ ⋃ such that the characteristic of the ring f ( q ) is q . Finally, let be the class of rings of the form ⊕ q ∈Q f ( q ) for some ( f , Q ) in . A corrected version of Theorem 6.2 [2] is Theorem 1. Let R be a ring with the descending chain condition on left or right ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if only if R belong to .

Abstract: Let Mn denote the number 2" — 1, where n is a positive integer. In 1644 Marin Mersenne, a friend and collaborator of Descartes and Fermat, published a list of numbers of this form which he believed to be prime. The numbers Mn are therefore known as Mersenne numbers (the primes among them being Mersenne primes). If n is composite, i.e. if it is the product of two smaller numbers h and k, then Mn is also composite, having the factorisation

Abstract: Recently B. R. Santos conjectured that 12 is the largest integer n with the following property: (*) J If m [ 1, n J and n are relatively prime, then n + m is a prime number. Using deep numerical estimates of Rosser and Schoenfeld for the number 7r(x) of primes less than x, it is proved that the conjecture of Santos is true. The same result holds, if in addition it is assumed in (*) that m is a prime. The positive integers not greater than a given integer and coprime to it are called its totitives. It is well known that 30 is the largest integer with the property that all its totitives are prime. B. R. Santos [4] proved that there exists a largest integer n with the property (1) 1 1r(2n) 1r(n) holds for n > 12. In order to prove (1) we use the following estimates due to Rosser and Schoenfeld [2, Theorems 1 and 15]. (RS 1) ir(x) > f(x) Igx I ox for x > 59 (RS2) X for) g() lg fo x>1 log x \2logx/ Received May 18, 1978. AMS (MOS) subject classifications (1970). Primary 1OA25; Secondary 1OH15, 10-04.

Abstract: Fermat’s theorem says that if p is a prime number and a is any integer relatively prime to p, then ap−1 ≡1 (modp). In Chapter I-11 we gave a proof based on the fact that the set of invertible elements of ℤ p forms an abelian group of order p ™1. In this chapter we give another proof based on the binomial theorem. We begin by making a definition which relates to the fact that in ℤ p , [p] = 0.

Abstract: Can your junior high school or high school students write the prime factor ization of a composite number? If so, then there are two rules your students can fol low: one to find the number of divisors of a natural number, > 2, and the other to find the sum of the divisors of a natural number, > 2. First, let's carefully define what we mean by the prime factorization or prime-fac tored form of a natural number, > 2. The fundamental theorem of arithmetic states that every natural number greater than one is either prime or can be written uniquely as a product of prime numbers. Thus, for ex ample, since 2, 3, 5, 7, and 1 are prime, we can write th? following:

Abstract: Recently (3) Mazur proved that if N is a prime number such that some elliptic curve E over Q admits a Q-rational isogeny then N is one of 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67 or 163.