Prime Number Distribution

[skilliyay]

There appears to always be an even total of prime numbers between the powers of 4, eg:

Between 0-4 TWO primes
Between 4-16 FOUR primes
Between 16-64 TWELVE primes
Between 64-256 THIRTY SIX primes
Between 256-1024 118 primes

I've tested the rule for the first 10 or so sets of primes and its always an even number, could anyone explain why that might be the case?

Thanks, Trev

[Canary42]

There's a challenge - someone prove Trev's Theorem - a la Fermat's Last Theorem which remained unproven for years - and I couldn't follow the eventual proof anyway ;-)

[fiction-factory]

Fascinating. I continued the pattern for a bit
1024-4096: 392 primes
4096-16384: 1336 primes

For the sake of accuracy if this is for work you are submitting, 4 to the power 0 is of course 1 not 0, but that doesn't change the results.

Sadly in my sleep I found myself working out differences between squares, and considering how primes tending to come in pairs, and formulated an idea- but I woke up with a headache and couldn't recall all the details.

[skilliyay]

Mathematical proofs go way over my head to Canary! I was speculating that someone may have a simple rationale for it

Thanks Fiction, don't give up, you might find something!

[Old_Geezer]

The reason is simple.
God decided to give you something to ponder about.

[Gizmonster]

According to the following, it doesn't hold for higher powers:

https://oeis.org/A117757

After 1336, the numbers are:

4642, 16458, 59025 (NOT EVEN), 213922, 781924, 2879938 ....... a few more odd ones later on as well ....

[skilliyay]

Wow thanks Giz, Primes always seem to throw up patterns that eventually break down at higher numbers, I guess that's why they can be so elusive....

[Gizmonster]

Thanks for the BA, but I will admit it wasn't my own work .... although I don't mind basking in the glory lol :)

[beso]

If you like freaky maths, read up on Heegner Numbers.

https://en.wikipedia.org/wiki/Heegner_number

[fiction-factory]

There is also the Goldbach conjecture which seems simple on the face of it but hasn't been proven as far as I know. This says that every even number above 2 is the sum of two prime numbers