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We prove that there are infinitely many integers n such that the total number of prime factors of (n+h1)…(n+hκ) is exactly (1 + o(1))κ Log k. Our result even ensures us that these prime factors are fairly evenly distributed among and factor n+hi. [ABSTRACT FROM PUBLISHER]

We demonstrate that, under suitable local conditions on a finite collection F1,…, Fg of binary irreducible forms with integer coefficients, the product F1(x)·…·Fg(x) will have at most r prime factors for infinitely many x. We give explicit upper bounds for r that depend only on g and on the total degree of the product polynomial. [ABSTRACT FROM PUBLISHER]

Let p be a prime and let J1(pn) denote the Jacobian of the modular curve X1(pn). The Jacobian J1(pn) contains a ℚ-rational torsion subgroup generated by the cuspidal divisor classes [(a/pn)−(∞)], where p∤ a. In this paper, we determine the structure of the p-primary subgroup of this ℚ-rational torsion subgroup in the case where p is a regular prime. [ABSTRACT FROM PUBLISHER]

The largest prime factor of X3+2 was investigated in 1978 by Hooley, who gave a conditional proo that it is infinitely often at least as large as X1+δ, with a certain positive constant δ. It is trivial to obtain such a result with δ=0. One may think of Hooley's result as an approximation to the conjecture that X3+2 is infinitely often prime. The condition required by Hooley, his R* conjecture, gives a non-trivial bound for short Ramanujan–Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of X3+2 is infinitely often at least as large as X1+δ, though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non-trivial bound for short Ramanujan–Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus. 2000 Mathematics Subject Classification: 11N32. [ABSTRACT FROM PUBLISHER]

As is usual in prime number theory, write $$\psi(x,q,a)=\sum_{{n\le x}\atop{n\equiv a\pmod q}}\Lambda(n).$$ It is well known that when q is close to x the average value of $$V(x,q)=\sum_{{a=1}\atop{(a,q)=1}}^q \left|\psi(x,q,a)-{x\over{\phi(q)}}\right|^2$$ is about xlog q, and recently Friedlander and Goldston have shown that if $$U(x,q)=x\log q-x\left(\gamma+\log 2\pi+\sum_{p|q}\frac{\log p}{p-1}\right),$$ then the first moment of V(x,q)-U(x,q) is small. In this memoir it is shown that the same is true for all moments. 2000 Mathematics Subject Classification: 11N13. [ABSTRACT FROM PUBLISHER]

To design a reliable system, it is very important to protect data stored or transmitted from intentional forgery in communications network. As a practical means of protecting data or software, it is pointed out by Naor and Yung that a family of one-way hash functions is useful and can be classified into a universal one-way hash function family (UOHF) and a collision intractable one-way hash function family (CIHF). The construction for a UOHF is possible under a weak assumption, i.e., one-way functions exist. However, the construction for a CIHF is not known to be possible under such a weak assumption. This paper presents a simple construction for a CIHF that is based on the hardness of factoring large composite numbers. The idea on the construction here is completely different from those reported so far and can be expected to provide a theoretical breakthrough to the construction for a CIHF under a general assumption. [ABSTRACT FROM AUTHOR]

Recently, many applications of integer theory to cryptographic techniques have been observed. The discrete logarithm problem is one such case. Usually, the discrete logarithm problem is the determination of the logarithm for the given arbitrary element with a prime number as the modulus. However, the discrete logarithm problem can also be considered with a composite number as the modulus. It is anticipated that the discrete logarithm problem with a composite number as the modulus is a difficult problem if the prime factors of the composite number, which is used as the modulus, are unknown. Then the problem can be applied to the cryptography. In the general discrete logarithm problem with a composite number as the modulus, it is not always true that an arbitrary element has a logarithm. From such a viewpoint, this paper shows that the exponent of an arbitrary element belonging to the irreducible residue class with a composite number as the modulus has a logarithm. Then the necessary condition in the application to the cryptographic technique is presented. Finally, as an application example of the technique proposed in this paper, a cryptographic technique based on the discrete logarithm problem with the composite number as the modulus is shown. [ABSTRACT FROM AUTHOR]

We examine the commutative rings for which every proper ideal has a proper prime factor. Among this class of rings, we classify the rings for which every proper ideal is a product of invertible and prime ideals. This continues and corrects a classification begun in an earlier paper. [ABSTRACT FROM AUTHOR]

The following two problems are open. Do two sets of positive integers [script A] and ℬ exist, with at least two elements each, such that [script A]+ℬ coincides with the set of primes ℘ for sufficiently large elements? Let [script A]={6, 12, 18}. Is there an infinite set ℬ of positive integers such that [script A]ℬ+1⊂℘? A positive answer would imply that there are infinitely many Carmichael numbers with three prime factors. In this paper we prove the multiplicative analogue of the first problem, namely that there are no two sets [script A] and ℬ, with at least two elements each, such that the product [script A]ℬ coincides with any additively shifted copy ℘+c of the set of primes for sufficiently large elements. We also prove that shifted copies of sets of integers that are generated by certain subsets of the primes cannot be multiplicatively decomposed. [ABSTRACT FROM AUTHOR]