Dear Norman Bradley Fox ,

The smallest n that satisfies your request is n = 4.

Your request is: the minimum n such that the interval [2ⁿ⁻¹, 2ⁿ]

includes at least 2n/2 - 2 primes.

I assume that n should be even, otherwise the number 2n/2 - 2 is not an integer

and you should replace it by the floor ( or ceiling) function!

n=1, [1,2] includes the prime 2

n = 2, [2,2²] includes the primes 2 and 3

n = 3, [2²,2³] includes the primes 5 and 7

n = 4, [2³,2⁴] includes 11 and 13 it includes exactly 24/2 - 2 = 2

primes which can be accepted.

To study the general case for your question, you can read more about Bertrand's postulate and its generalizations that discuss the number of primes in the interval

[m, 2m].

https://en.wikipedia.org › wiki › Bertrand's_postulate

www.lolathompson.com › uploads › bertrands_postulate

I've read up on some generalizations on Bertrand's Postulate, but I'm still not coming up with a proof of why there is a guarantee of 2^(n/2)-2 primes in [2^(n-1),2^n]. The closest I've seen is reasoning of why there is at least sqrt(2n)/n primes in (n,2n), which would give me 2^(n/2)/2 for my interval, so not quite to the result I need.

Norman Bradley Fox ,

Your conjecture that

There are 2n/2 - 2 primes in the interval [2^(n-1), 2^n], in general, is not correct.

Counterexample:

Choose n = 6

[2⁴, 2⁵] = [16, 32] includes the following five prime numbers {17, 19, 23, 29, 31}

which is less than the conjectured number

2n/2 - 2 = 26/2 - 2 = 8 - 2 = 6.

For n=6, it would be the interval [2^5,2^6], which would include 37, 41, 43, 47, 53, 59, and 61, so this would exceed 2^3-2=6. I was not claiming in my question that this is true for all n (and I did forget to state n was even). I am trying to determine if it is true for all n>N for some positive integer N and to figure out why the inequality would be true for those cases.

Norman Bradley Fox ,

In this case, examples will show nothing, theoretical proof is a must.

We can restate your conjecture as to the following:

There exists k ∈ N such that [22n-1, 22n] includes at least ( 2n - 2 ) prime numbers for every n ≥ k.

Can you state an evident that makes you feel this conjecture is true?

I did state the question poorly because I meant an integer N so that for all n>N, there exists....

I think I have determined why my conjecture is true now based on the upper and lower bounds of pi(n) coming from the Prime Number Theorem.