Search

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]

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 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.