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 a1,a2,a3,...,andenote digits and not multiplication):

a1a2a3...an1, a1a2a3...an2, a1a2a3...an3, a1a2a3...an4, a1a2a3...an5, a1a2a3...an6, a1a2a3...an7, a1a2a3...an8, a1a2a3...an9, a1a2a3...(an+1)0

The entire list of consecutive natural numbers I showed above can be made composite if:

- The number formed by digits a1a2a3...an should be a multiple of 3
- The numbers a1a2a3...an1 and a1a2a3...an7 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.

**Answer**

You can have a sequence as long as you wish. Consider n∈N then the set

Sn={n!+2,n!+3,⋯,n!+n}

is made of composite consecutive numbers and is of length n−1

**Attribution***Source : Link , Question Author : Pritt Balagopal , Answer Author : marwalix*