The Holy Grail Of Mathematics.

[Don Blazys]

To: The Big Dog.

 

The key (second) root of the equation actually must be transcendental in order to generate primes, and so the task of determining its exact value will indeed be equal to, and perhaps even much more difficult than the determination of the primes themselves. The sequence of primes appears to be utterly unpredictable, so the real challenge here is to at least determine whether or not there even exists some number such as;

 

2.566543832171388844467529...

 

that can possibly be defined by some equation, series, or algorithm, because that would imply an underlying predictability.

 

There are also many, many other "Blazys constants" such as:

 

2.313036736... and 2.794019072...

 

that also generate the entire sequence of primes, in order of magnitude by applying different methods and/or rules. (In the above two examples, we simply subtract the whole number part, then take the reciprocal of the result.)

 

Proving that all such "Blazys constants" are not definable is probably impossible, so all we are left with is the possibility of trying "this, that and the other thing", just to see how far into the primes some particular "possible definition" of a "Blazys constant" will take us.

 

The formula in my previous post happens to generate all the primes up to and including 59, and is therefore, at present, the "world record" for a "true prime generating function". (One that does not involve sieves, comparisons or "Wilsons theorem").

 

At present, all known formulas such as the one in this thread are merely "curiosities" or "curios", but as I happen to take great delight in discovering them, I will continue to try "this, that and the other thing", in my spare time, if only for the fun of it.

 

As I said before, the probability that such a formula even exists is quite likely to be zero, but if by some miracle a really promising one is ever found, then it will indeed represent a major breakthrough.

 

Before I discovered the "cohesive terms" that are discussed in the thread "A Mathematical Emergency", I really didn't believe that it was algebraically possible to prevent cancelled variables from being "crossed out", or in any such "mathematical miracle".

 

Now I do.

 

Don.

[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. 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.

[Don Blazys]

To: Nootropic,

 

When this formula was first discovered, the empirical evidence seemed very convincing. Since then, it was discovered that the second root of the original equation actually departs from the prime generating "Blazys constant":

 

2.566543832171388844467529...

 

at the thirteenth digit while the second root of the revised equation from several posts ago deviates from the "Blazys constant" at the twenty-sixth digit.

 

Sorry, but there is no "derivation" in that this formula was never derived from any well established principles, but was instead based on what, at the time, appeared to be a very remarkable coincidence between two seemingly unrelated expressions. (From time to time, albeit very rarely, this sort of thing does indeed occur in mathematics, so I felt compelled to share it.)

 

Thus, this formula really is a "frankenfunction" (what a clever word!), and does indeed come from "nowhere".

 

That said, it is still the most succsesfull true prime number generating function ever discovered and a remarkable "curio" to boot! (To fully appreciate the degree of difficulty involved, try finding a formula that generates just the first 10 primes in order of magnitude without relying on seives, comparisons or Wilson's theorem.)

 

Personally, I suspect that the paradigm of defining a "Blazys constant" via some equation, series or algorithm is really the only chance we've got for developing a true prime generating function. However, I'm not expecting another miracle.

 

Please keep in mind that this is only my "fun thread" (the words "Holy Grail" strongly imply "something that is probably fictional") that I began in order to to offset my far more serious thread entitled "A Mathematical Emergency".

 

Don.

[CraigD]

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 [math]A_1[/math] (the "Blayze contant") such that for the sequence

 

[math]A_n = \frac1{\frac{A_{n-1}}{\lfloor A_{n-1} \rfloor}-1}[/math]

 

generates the primes [math]\{ P_1, P_2, \dots P_n = \lfloor A_n \rfloor \}[/math]

 

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 [math]A_1[/math] doesn’t exist. I showed that, if the first 5 primes were 2, 3, 5, 7, 23 rather than 2, 3, 5, 7, 11, [math]A_1[/math] 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 [math]A_1[/math] 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 [math]A_1[/math]. 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 [math]A_1[/math], as you must know all of the primes to evaluate it.

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 [math]A_1[/math] 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 [math]A_1[/math] 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 [math]A_1[/math] 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: [math]\frac{5355306055405720589112}{2086582737562264077801}[/math]

 

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.

 

[math]\frac{5355306055405720589112}{2086582737562264077801}[/math] 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 .

[billnye]

If i am correct what you are suggesting is the riemann hypothesis? here is a link that i read about after i read this forum: Maths holy grail could bring disaster for internet | Technology | The Guardian

 

This explains that such a proof or solution could destroy internet commerce because it would effectively be the key in decrypting any bank or credit card information. It also says that this could cause chaos basically if used wrongly. i suggest if that is what you have found that you contact the organization defined on that site: the clay mathematics institute and it also says that there would be a $1 million prize. It sounds to me that this could be very dangerous in the wrong hands so please use your discretion.

[billnye]

^I am only 16 so forgive me if i am wrong.^

[Don Blazys]

To: billnye,

 

You are right.

 

But unfortunately, I was wrong.

 

The formula looked like it would work,

but turned out to be just a really wierd and incredible coincidence.

 

A mere "curio".

 

I now suspect (but can't prove),

that all such formulas work only to a point, then "break down".

 

Don.