Answer by barak manos for Sets of Prime and Composite Numbers
It is very simple to construct an infinite sequence for the Prime-Composite case:$$23+60n,25+60n$$$25+60n$ will generate an infinite amount of composite numbers, all of which are divisible by $5$ (in...
View ArticleAnswer by Dan Brumleve for Sets of Prime and Composite Numbers
Case $2$ is answered so I will address the others. For case $3$, by Dirichlet's theorem there are an infinite number of primes of the form $10 \cdot n + 3$, and $10 \cdot n + 5$ is always divisible by...
View ArticleAnswer by David for Sets of Prime and Composite Numbers
The composite-composite case is easy. By the Chinese remainder theorem there are infinitely many solutions of, for example,$$x\equiv0\pmod6\ ,\quad x\equiv1\pmod5\ ,\quad x\equiv-1\pmod7\ .$$And for...
View ArticleSets of Prime and Composite Numbers
We know that all primes are of the form $ 6k ± 1 $ with the exception of 2 and 3.We also know that not all numbers of the form $ 6k ± 1 $ are prime.This leads to four distinct sets of pairs adjacent to...
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