A number theory problem by Achal Jain
Does there exist a natural number n such that the numbers n + 1 , n + 2 , n + 3 , … , n + 1 9 8 9 are all composite?
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2 solutions
This statement, in its present form, does not provide the solution to the problem in my opinion. The largest prime gap known to date is not as small as 1 3 6 9 2 , that would be ludicrous in this computer age. The correct statement is that the largest known maximal gap has length 1 3 6 9 2 ( The gap g n is called maximal if if g m < g n for all m < n .
Plus, you don't have to know anything about known maximal gaps to solve this problem. A better solution is given by @Sam Bealing by explicitly stating such n .
Let n = 1 9 9 0 ! + 1 then 2 ∣ n + 1 = 1 9 9 0 ! + 2 as 2 ∣ 1 9 9 0 ! and so on up to 1 9 9 0 ∣ n + 1 9 8 9 .
Doesn't n = 1 9 8 9 ! suffice?
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No because we can't say a number divides n + 1 = 1 9 8 9 ! + 1 .
As of June 2016, the largest prime gap -- the difference between two prime numbers -- as discovered by D. Jacobsen, is 13692. The numbers within the prime gap are composites.