Is that supposed to be a joke ? Cos you can easily calculate A, like the 5th prime number is 7 so by setting A^(3^5) =7, you can workout A to be 1.0080…
It’s f(n) = ⌊A[sup]3[sup]n[/sup][/sup]⌋, where n is a positive integer and it’s not a formula for the nth prime, just a function whose values are all prime numbers.
Man - that is still not possible, i think. I am not a mathematician, so maybe someone can present an elegant proof. Setting n=1, gives A to be a cube root of a prime number say P1. So - by definition all subsequent values of the functions will be divisible by P1 - am I missing something ?
What you’re missing is that f(x) = ⌊x⌋ denotes the floor function which rounds down to the nearest whole integer. E.g. ⌊2.67775⌋ = 2.
As Asympotically fat states, you’re missing the ⌊ … ⌋ marks, which mean A is only approximately the cube root of a prime …
Here’s a proof. It’s very short and I think I almost understand it! 
I’ve a question about Wilson’s Theorem. The related Fermat’s little theorem has a very elegant combinatorial proof. Does Wilson’s Theorem have a similarly elegant proof? If not, how many kilohours have been spent looking for one?
Oh and I was mistaken that no one has a clue what it is. IF Reimann’s Hypothesis is true then we know what it is.
Well don’t hold back on us!
The constant Mills discovered is
Mills’ Constant
Thanks, that’s enough to Google on. It looks like the (presumed) value is approximately 1.3063778838630806904686144926…, with no known neat way of calculating that number.
The way I read the Wiki article is that Mills number is the smallest value that A in the above could take on if the R. hypothesis is true (FWIW basic arithmetic tells me that A cannot be any smaller than approximately 1.26). I’m guessing that by knowing each successive prime calculated by the above formula when A is Mills number allows you to arrive at a more accurate value for Mills number.
I read it that there is known to be some smallest number that fits the bill, and that that smallest number (whatever it is) is defined to be Mills’ Constant, and that 1.306 etc. is the value if the Riemann hypothesis is true.
My favorite proof that the number of primes is infinite is if you take any two consecutive primes (call them pm and pn), multiply them and add one, the result is a prime. I like it because it’s pretty easy to understand (small minds think alike).
However, this doesn’t give us a prime number generator, because we won’t know the next lower consecutive prime from the newly generated one, And it certainly doesn’t generate them all.
That’s a mis-statement of Euclid’s proof and factually inaccurate.
Counter-examples abound, because consecutive primes, except for 2 and 3, will both be odd numbers. Multiply any 2 odd numbers together, add 1, and you get an even number greater than 2.
For example: 1113 + 1 = 143+1 = 144 = 272
In fact, not only is it wrong, it’s very nearly maximally wrong.
Even for large values of 1?
Thanks! Obviously I misrembered this; I’ll have to rethink.
Even for very large values of 1.
:smack:
Giggle.