Hello and I'm quite new to Math SE.
I am trying to find the largest consecutive sequence of composite numbers. The largest I know is:
$$90, 91, 92, 93, 94, 95, 96$$
I can't make this series any longer because $97$ is prime unfortunately.
I can however, see a certain relation, if suppose we take the numbers like (let $a_1, a_2, a_3,…,a_n$denote digits and not multiplication):
$$a_1a_2a_3…a_n1,\ a_1a_2a_3…a_n2,\ a_1a_2a_3…a_n3,\ a_1a_2a_3…a_n4,\ a_1a_2a_3…a_n5,\ a_1a_2a_3…a_n6,\ a_1a_2a_3…a_n7,\ a_1a_2a_3…a_n8,\ a_1a_2a_3…a_n9,\ a_1a_2a_3…(a_n+1)0$$
The entire list of consecutive natural numbers I showed above can be made composite if:
- The number formed by digits $a_1a_2a_3…a_n$ should be a multiple of 3
- The numbers $a_1a_2a_3…a_n1$ and $a_1a_2a_3…a_n7$ should be composite numbers
If I didn't clearly convey what I'm trying to say, I mean like, say I want the two numbers (eg: ($121$, $127$) or ($151$, $157$) or ($181$, $187$)) to be both composite.
I'm still quite not equipped with enough knowledge to identify if a random large number is prime or not, so I believe you guys at Math SE can help me out.
Best Answer
You can have a sequence as long as you wish. Consider $n\in\Bbb{N}$ then the set
$$S_n=\{n!+2,n!+3,\cdots,n!+n\}$$
is made of composite consecutive numbers and is of length $n-1$