Science Forums 2

[CraigD]

Originally Posted by Nootropic

I've glanced over the previous pages of this thread, and someone correct me if I'm wrong, but nowhere did I see a proof that this sequence generates the entire sequence of prime numbers.

In post #60, I speculated that the existence of an initial constant (the "Blayze contant") such that for the sequence



generates the primes

is equivalent to the Riemann hypothesis’s strengthening of the prime number theorem, because if an unexpectedly large gap in the primes exists, it can be shown that doesn’t exist. I showed that, if the first 5 primes were 2, 3, 5, 7, 23 rather than 2, 3, 5, 7, 11, would not exist.

I haven’t attempted to prove this equivalence, though, and suspect that such exceeds my skill.

So, in short, we’re only speculating that exists, with about the same confidence that we speculate that the Riemann hypothesis is correct.

I’ve been trying to find a terse expression for the value of . At present, I’ve the program in post #60, but suspect there’s a more efficient program that can be tersely described as an infinite series. Note that there’s no “magic” to such an expression for , as you must know all of the primes to evaluate it.

Originally Posted by Nootropic

Quite honestly, you can give all the empirical evidence you want, but it's relatively useless regarding your claim. And the formula really has the feeling that it "comes from nowhere" and it looks like some kind of pieced together frankenfunction that calculus students were asked to differentiate. What I'm saying is that I would really like to see the derivation of said formula.

I think Don’s approach to finding a finite expression for is best described as a “search for an amazing coincidence”. The hope that such a coincidence can be found stems, I think, from the common “mathematical superstition” that many transcendental numbers are subtly related to one another, so a transcendental number generated from all of the primes might be related to an expression of some “famous” constants and/or functions.

Proving this not to be true seems challenging. I’m fairly sure that a proof that is not rational could be based on any expression/program to generate it, and wouldn't be very difficult.

My main interest in numbers like this have to do with viewing them as data compression algorithms. Consider, for example, Don’s latest expression for an approximation of which generated the first 17 primes, failing on the 18th:
(((e^(e^(((((((((e^(pi))*((2*3*4-1)*((2+3+4)^2-2)+1/x))^(-1)+1)^(-1)*e^(x+2*3*5-1))^(-1)+1)*sin(x^(1/2)))^(-1)-1)^(-1)-(pi)^2)^(-1)))/2-1)^(-1)/2-1)^(-1)/x-1)=0

a rational number that does the same:

and the first 17 primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

Depending on how they’re represented (assuming it’s reasonably efficient), the first 17 primes require between 126 to 160 bits.

requires 156 to 229 bits.

Don’s expression requires from 477 to 734 bits, much less information dense than the first 17 primes. However, if Don could find a short expression that generated thousands of primes, it would be more information dense than the generate primes, while a rational approximation would likely be about the same.

From a data compression point of view, generating the primes isn’t all that interesting, because, represented efficiently, a simple algorithm such as:
B= 2;
do while B >1 { B=A; A=A+1;
do while B>1 and A mod B = 0 { B= B-1}
}
requires about 188 bits, and will (very inefficiently) generate all of the primes. What’s more interesting to me are terse expressions that generate very long finite sequences – see [thread]“The Starburst Challenge”[/thread].