I have been working on a problem as follows:
Do there exist 100 consecutive natural numbers none of which is prime?
I know that the answer is 'yes', by considering 101!, and noting the sequence 101! + 2, 101! + 3, ... , 101! + 101.
This approach generalises nicely by considering (n+1)!
However, whilst tacking this problem, I tried many different techniques.
The approach I was most interested in was the following intuition:
We know that the primes are much more spread out than occurring every n integers from knowledge beyond the problem. Since every number can be factorised uniquely into primes despite primes being rare, if we have a prime at least every 100 numbers, say, then we should surely be able to show that the number of possible combinations of primes exceeds the number of numbers that exist.
My problem is that I have found it hard to count the total number of combinations.
For example,
Say a prime occurs at least once every 100 numbers. Then there are at least N primes less than 100N. How many prime combinations can you make that are less than 100N? I'm hoping that I can get a result that exceeds 100N, and therefore we show that the primes cannot populate the natural numbers this densely.
Sorry for the long question! Just thought I'd give some background to my question.
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