The Holy Grail Of Mathematics.

[CraigD]

Originally Posted by modest

Originally Posted by Qfwfq

Wouldn't it be simpler to write as follows? (I don't know what's gone wrong with the width here)

You’ve skipped on of the -1 exponents. It should be

[pgrmdave]

I'm not great with math, but this is one of the coolest things I've ever seen

It almost seems to me to be a way to transcend our current numerical system, by being able to define all numbers using universal constants, sort of like a Grand Unified Theory for mathematics.

[Qfwfq]

Originally Posted by CraigD

You’ve skipped on of the -1 exponents.

I'm not so sure of that. I suspect you've counted one of them in two different ways.

Remember that the reciprocal of 1 is 1.

[Don Blazys]

To: Modest, CraigD, Qfwfq, Pgrmdave and everyone else who participated in the "search for the Holy Grail of mathematics".

Eliminating the logarithms by re-writing the equation as:

((e^(e^((((sin(x^(1/2)))^(-1)-1)^(-1)-(pi)^2)^(-1)))/2-1)^(-1)/2-1)^(-1)/x-1=0,

I was able to confirm Modest's results with the online calculator that he recommended.

I thank each and every one of you for all your help and hard work.

I know that some of you are dissapointed in that the equation provided us with a remarkable approximation rather than an exact value for the "Blazys constant":

2.566543832171388844467529... ,

but please let me assure you that I am probably more disappointed than you.

Besides, I have several dozen similar equations that are also good "candidates", so in reality, the quest has just begun! I will be checking them over during the Christmas holidays, and will keep you posted as to how they pan out.

You know, since the equation on my website generates the primes:

(2, 3, 5, 7, 11, 13, 17, 19, 23 and 29)

in that order and without involving "sieves", "comparisons" or "Wilsons theorem", it is still the best and most promising "prime generating formula" to date!

In fact, a mathematician in the Marilyn vos Savant forum recently expressed enormous surprise that there is nothing even remotely resembling it in "Wikipedia", and my guess is that he and others will now join me in the search for an equation, expression or series that results in an exact value for my constant.

Don.

[modest]

You're very welcome, Don. Keep us updated.

~modest

[Nootropic]

http://www.scottaaronson.com/blog/?p=304

I believe everyone should have a gander at these, they are important.

[Don Blazys]

"Tweaking" the original equation, I was able to garner a substantial improvement in the accuracy of the prime generating constant:

2.566543832171388844467529...

The second root (zero) of the equation:

(((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

is:

2.566543832171388844467529401576979616943754028928 2...

which is accurate to 25 decimal places and generates the primes:

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

in order of magnitude.

I am now convinced that given enough time, we could continue to "tweak" this (and many similar equations) so as to generate as many primes, in order of magnitude, as we so desire.

Although an exact result may very well be impossible, I have decided that this is a very unique endeavor, (and a lot of fun,) so from time to time, I will continue to post improvements in accuracy. After all, this thread is called the "Holy Grail Of Mathematics"....and you just never know...someone might even spot an interesting pattern!

I'm going on vacation now, and won't be near a computer for a week or so.

I wish you all a very merry Christmas and a happy Hypographical New Year.

God bless you!

Don.

[TheBigDog]

I have been following this at a bit of a distance. I would speculate that the key root of the equation that you are seeking will in fact turn out to be a infinite non-repeating decimal similar to pi, with greater and greater precision required to keep the accuracy as the primes get larger. The task of finding the decimals of this root will equal the task of seeking the primes themselves. Have you considered this possibility?

Bill

[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.