Infinite Gaps Between Prime Numbers

In this video, we are going to show that for any given positive n, we always can find a sequence of numbers of length n such that none of these numbers is a prime number.

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Please watch: "Real Projective Space, n=1"
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[xaxuser]

We can use any k! For example with the case of 100 we have that 100!+1 is not prime and we also have 100!+2 not prime and 100!+3 and to 100!+99 and 100!+100 all this numbers are composite integers and they are successive so we find a sequence of numbers that have no primes in betwen with gap of 100
And we can try it for any huge number k to find any gap that we want

[SiggiTheHopper]

I get why you can find a sequence of any length of non-prime numbers, but if you are interested in a sequence of a given length, say 100, why are a_1 -1 and a_100 +1 prime numbers? Because, if they are not, then you only have a sequence of bigger length, but it´s not clear that there exists such a sequence with length of exactly 100.

[ivanmasmejor]

But I saw something I think is wrong. The set which has not prime numbers needs to into prime numbers, but if i=101 (so u already have 100 nonprime numbers) if you put into de definition of the set u are going to have (2*3*4*5*...*100*101+102) and it is not prime because the gcd(101!, 102)>1 so u are going to have more than 100 nonprime numbers between 101!+1 (the prime number before having 100 nonprime numbers) and the firts prime wich follows the set. I think that number is 101!+103 because gcm(101!, 103)=1 and by that reason u are going to have 102 nonprime numbers between 101!+1 and 101!+103