A variant of the Hardy-Ramanujan theorem

For each natural number $n$, we define $\omega^*(n)$ to be the number of primes $p$ such that $p-1$ divides $n$. We show that in contrast to the Hardy-Ramanujan theorem which asserts that the number $\omega(n)$ of prime divisors of $n$ has a normal order $\log\log n$, the function $\omega^*(n)$ does not have a normal order. We conjecture that for some positive constant $C$, $$\sum_{n\leq x} \omega^*(n)^2 \sim Cx(\log x). $$ Another conjecture related to this function emerges, which seems to be of independent interest. More precisely, we conjecture that for some constant $C>0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.

Small Prime Divisors Attack and Countermeasure against the RSA-OTP Algorithm

One-time password algorithms are widely used in digital services to improve security. However, many such solutions use a constant secret key to encrypt (process) one-time plaintexts. A paradigm shift from constant to one-time keys could introduce tangible benefits to the application security field. This paper analyzes a one-time password concept for the Rivest–Shamir–Adleman algorithm, in which each key element is hidden, and the value of the modulus is changed after each encryption attempt. The difference between successive moduli is exchanged between communication sides via an unsecure channel. Analysis shows that such an approach is not secure. Moreover, determining the one-time password element (Rivest–Shamir–Adleman modulus) can be straightforward. A countermeasure for the analyzed algorithm is proposed.

N-critical graph of finite groups

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.

On closed classes in partial k-valued logic that contain all polynomials

Let Pol k be the set of all functions of k-valued logic representable by a polynomial modulo k, and let Int (Pol k ) be the family of all closed classes (with respect to superposition) in the partial k-valued logic containing Pol k and consisting only of functions extendable to some function from Pol k . Previously the author showed that if k is the product of two different primes, then the family Int (Pol k ) consists of 7 closed classes. In this paper, it is proved that if k has at least 3 different prime divisors, then the family Int (Pol k ) contains an infinitely decreasing (with respect to inclusion) chain of different closed classes.

Finding all S-Diophantine quadruples for a fixed set of primes S

Given a finite set of primes S and an m-tuple $$(a_1,\ldots ,a_m)$$ ( a 1 , … , a m ) of positive, distinct integers we call the m-tuple S-Diophantine, if for each $$1\le i < j\le m$$ 1 ≤ i < j ≤ m the quantity $$a_ia_j+1$$ a i a j + 1 has prime divisors coming only from the set S. For a given set S we give a practical algorithm to find all S-Diophantine quadruples, provided that $$|S|=3$$ | S | = 3 .

A note on weakly prime-additive numbers

A positive integer [Formula: see text] is called weakly prime-additive if [Formula: see text] has at least two distinct prime divisors and there exist distinct prime divisors [Formula: see text] of [Formula: see text] and positive integers [Formula: see text] such that [Formula: see text]. It is easy to see that [Formula: see text]. In this paper, intrigued by De Koninck and Luca’s work, we further determine all weakly prime-additive numbers [Formula: see text] such that [Formula: see text], where [Formula: see text] are distinct odd prime factors of [Formula: see text].

On a bound of Cocke and Venkataraman

Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of $$\gcd (m,4)$$ gcd ( m , 4 ) and the odd prime divisors of m. We show that $$|G|\le q(m)k^2/\varphi (m)$$ | G | ≤ q ( m ) k 2 / φ ( m ) where $$\varphi $$ φ denotes Euler’s totient function. This strengthens a recent result of Cocke and Venkataraman. As an application we classify all finite groups with $$k<36$$ k < 36 . This is motivated by a conjecture of Thompson and unifies several partial results in the literature.