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The function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to the first k primes. We give a new computational method for calculating strong upper bounds on h(k). [ABSTRACT FROM AUTHOR]

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p - 1 divides a highly composite A [ABSTRACT FROM AUTHOR]

The article discusses the equation for the number of positive divisors of the natural number. Noted that a natural number that is greater than 1 is harmonic provided that the number of positive divisors of the natural number and its sums, has prime factor decomposition. Illustrated is the formula of a natural number with multiplicative functions. In addition, two theorems are detailed concerning conditions wherein a natural number is lesser and/or equal to the cube root of a natural number.

In this paper, we find all integers $ x$ has only prime factors smaller than $ 100$ we can find the largest positive integer $ x$ $ x, x+1,\ldots, x+n$ [ABSTRACT FROM AUTHOR]

The article presents applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. It examines the Pollard's algorithm which shows that prime factors can be recovered in deterministic polynomial time. Moreover, it also notes on how to compute Euler's totient function, the Extended Riemann Hypothesis and the complexity of factoring given an oracle for the function.

The article presents a mathematical analysis that examines the method for computing prime harmonic sums. It details the method of the study that considers the Meissel-Lehmer algorithm in computing the prime-counting function, which is improved by several mathematicians. The findings of the study reveal the effectiveness of the algorithm in determining the computation of prime harmonic sum.

The article focuses on a new algorithm that can compute the f (k) function, which is introduced by Paul Erdős to measure the sequence integers minimal gap size with at least one prime factor that exceeds k. It says that the algorithm shows that f (k) is not a monotone. It also states that the computations that led to the Proposition 3.1 is run and split on a Mac Power personal computer (PC) dual core gigahertz (GHz) processor.

In this paper we construct a cover ${a_{s}(operatorname {mod} n_{s})}_{s=1}^{k}$ of $mathbb {Z} $ with odd moduli such that there are distinct primes $p_{1},ldots ,p_{k}$ dividing $2^{n_{1}}-1,ldots ,2^{n_{k}}-1$ respectively. Using this cover we show that for any positive integer $m$ divisible by none of $3, 5, 7, 11, 13$ there exists an infinite arithmetic progression of positive odd integers the $m$th powers of whose terms are never of the form $2^{n}pm p^{a}$ with $a,nin {0,1,2,ldots }$ and $p$ a prime. We also construct another cover of $mathbb {Z} $ with odd moduli and use it to prove that $x^{2}-F_{3n}/2$ has at least two distinct prime factors whenever $nin {0,1,2,ldots }$ and $xeq a (operatorname {mod} M)$, where ${F_{i}}_{igeqslant 0}$ is the Fibonacci sequence, and $a$ and $M$ are suitable positive integers having 80 decimal digits. [ABSTRACT FROM AUTHOR]

The article discusses a mathematical strategy which extends the bounds on the total number of prime factors of an odd perfect number. It proves the theorem that if N is an odd perfect number, then the total number of prime factors of N is greater than or equal to 75 (Ω(N) ≥ 75). It mentions that the algorithm in Ω(N) ≥ 47 was modified to speed up the calculation and reduce storage space requirements.

The article presents an analysis on the theory that odd perfect number has at least nine distinct prime factors. It discusses the contradictions and propositions used for the analysis as well as the results used including the results of 16 for prime numbers not greater than 17 and the proof of non-computational lemmas. It says that factorization has been performed using probable primality test.

A correction to an article on the largest prime factor x2-1 published in a previous issue is presented.