The Holy Grail Of Mathematics.

[Don Blazys]

I have been doing mathematical research for over 40 years. In that time, I made only two major discoveries, along with a dozen or so that are "interesting", but not quite "major".

 

Now, one of my "major discoveries" (a new and very powerfull basic algebraic term that I named a "cohesive term") is already well documented on my website (donblazys.com), and in several forums (including this one). I call it a "major discovery" because I'm absolutely certain that it is the paradigm and foundation of a new, perfectly defined and self consistent form of algebra, that is infifitely more powerfull than the existing system. It is both easy, and mindblowing to see that "cohesive terms" have properties that are quite remarkable, such as their preventing any loss of cancelled common factors and preventing the existence of trivial common factors.

 

My other "major discovery" occured while I was researching the one and two term prime counting functions (Pi(x) approximation formulas) that you can find on my website (donblazys.com).

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It is a formula that generates the entire sequence of prime numbers, in order of magnitude, using only the constants e and Pi, and the single variable x.

_________________________________________________________________________________________

 

I have kept this formula to myself, (well, I did show it to my wife and son, but they are not mathematicians) because it might be used to compromise encryption and security codes. However I have always felt ambivalent and "torn" about that because this formula also represents the key to a much deeper understanding of prime numbers, and therefore requires further study by mathematicians other than myself.

 

Yesterday, I came up with a plan. Here it is:

 

Without actually showing the formula to anyone, I can demonstrate that it works by putting it into a brand new, unopened graphing calculator, and then handing that calculator over to some qualified mathematician who could then inspect the roots (zeroes of the graph) and verify that the formula does indeed work. That would eliminate the possibility that I'm just a delusional "crank". That "qualified mathematician" could then disseminate the fact that such a formula does indeed exist, and I would then be able to publish my prime generating formula in this and other forums after enough time has been given to all users of prime based encryption to take the necessary precautionary measures.

 

When I told my wife of this plan, she got angry with me for wanting to "give the formula away", rather than sell it! She told me to think of the grandkids, and how the extra money could be used to further their education.

 

Now, I'm really lost! Are there security companys or institutions of higher learning that would buy such a formula? Can I put it up for auction on e-bay? Can it be patented?

 

Then again, if I did sell it, the buyer would forever keep it a "trade secret" and the math community as a whole, would not get access to what I know is the Holy Grail of mathematics!

 

To all of the very kind and intelligent folks in this forum, I really need your heartfelt advice and informative input.

 

What would you do if you were me?

 

Don.

[Buffy]

What would you do if you were me?

Get a qualified intellectual property/patent attorney.

 

If its as good as you say it is, the amount you spend on legal fees will be spit in the ocean.

 

If you do anything else, you're going to find that your wife is right....

 

You must lose a fly to catch a trout, :phones:

Buffy

[Don Blazys]

To: Buffy,

 

Thanks for the advice.

 

Getting an intellectual property/patent attorney probably is my first order of business, but you make it sound like I won the lottery!

 

How and why can this formula make a lot of money?

 

Where would the money come from?

 

Would it be possible to make money from it and also make it public so that it, along with its consequences and ramifications can be studied?

 

Don.

[Buffy]

That's what the person who will get the *other* half of the "fortune" will tell you when you find them and have the sign a non-disclosure agreement.

 

I would be concerned however that if you have no idea of what the consequences of your discovery are, that its highly likely that it quite possibly has no useful value!

 

It's also likely that you will not be able to successfully obtain a patent if you cannot enunciate any concrete applications of your discovery, and an IP attorney will tell you that the only hope you have is to maintain it as confidential and proprietary information as part of a business that uses it (kind of like Col. Sanders secret recipe).

 

Good ideas are a dime a dozen. It takes real ingenuity to figure out how to make money off of them....

 

Invention is the mother of necessity, :phones:

Buffy

[Essay]

Get a qualified intellectual property/patent attorney.

...sounds like the best advice....

===

 

Something like this seems as if it should reveal some significance in the world of dimensions and numbers.

I would be inclined to try and publish it in a mathematics journal and hope the recognition would lead to some lucrative pursuits. :phones:

 

;) So can you choose to generate a prime with any certain number of digits?

 

Science News / Largest Known Prime Number Found

Sunday, September 28th, 2008

Here’s a number to savor: 2^43,112,609 -1.

The Great Internet Mersenne Prime Search, or GIMPS, a computing project that uses volunteers’ computers to hunt for primes, found the prime and just confirmed the discovery. It can now claim a $100,000 prize from the Electronic Frontier Foundation for being the first to find a prime number that has more than 10 million digits.

Mister-Info.com - Two largest known prime numbers discovered just two weeks apart, one qualifies for $100k prize

Two largest known prime numbers discovered just two weeks apart, one qualifies for $100k prize

Two new records for the largest known prime number have been set, both breaking the 10 million digit threshold. On August 23, Edson Smith, a systems engineer for the Program in Computing laboratory at the University of California, Los Angeles in California, United States, confirmed the primality of the number through his work as a volunteer in the distributed computing project known as the Great Internet Mersenne Prime Search, or GIMPS. This new prime qualifies GIMPS for a $100,000 award from the Electronic Frontier Foundation (EFF), offered to the first person or group to discover a prime number of 10 million digits or more.

...perhaps hold on until there is a 100 million-digit prime prize?

 

~ ;)

[sanctus]

By publishing it in a mathematical journal you automatically get the property/patent of it, if I understood things right. So if it is really a such a great formula as you say, then you should do it.

because it might be used to compromise encryption and security codes

You should not have concerns about this, simply because you might be the first to have found that formula, but if you don't publish it someone else will find it in the future (1 year/10years...not really important) and publish it...or do you believe to be so unique that no1 else can go down the path you went to find that formula?

So write an article (if you have never written them, check out on arxiv or elsewhere how they are written) and submit it to a journal. Then some "qualified mathematician" will review it and tell you what you have to change to get it published...

By the way I just figured out a few days ago how it works when publishing an article. You send it in to a given journal, this journal then sends it to a researcher acive in that field (usually a professor I guess), who then gives feed-back directly to you.

[CraigD]

Post a list of the largest consecutive numbers you’ve generated with the algorithm, as many as you can within reason, perhaps 100 or 1000 numbers. Those of us who are good a high-speed arithmetic (using computers, of course) can check those numbers for primality, using whatever method he have handy. If any obvious composite numbers are discovered, you’ll know there’s a problem, and be able to investigate.

 

There’s no need to describe your algorithm with this approach, and checkers can simply assume you’re honest about using an unknown algorithm – that is, that you’re not using a known prime number generator or getting them from a published list generated by a known method, and bogusly claiming to be using something else.

 

It’s a waste of your effort to work on getting an IP lawyer or publishing in a journal if the algorithm fails upon simple examination, so simple examination is a worthwhile first step.

[Qfwfq]

I'd say Craig's idea is reasonable but to prove your case better it ought to be more like: you folks choose some high number and I list the primes that follow it. If your reply comes after a short time compared to known methods then we might be convinced that you are indeed employing a novel and quick method...

[Pyrotex]

Sanctus is right.

Typically, when any new idea is invented or discovered, it is because the background body of available knowledge has made the new idea possible to invent, or possible to be discovered. By anybody tapped into that available knowledge!!

 

So, your discovery is almost certain to be duplicated in the next one or two decades.

 

Publish now, or lose it.

[Turtle]

So write an article (if you have never written them, check out on arxiv or elsewhere how they are written) and submit it to a journal. Then some "qualified mathematician" will review it and tell you what you have to change to get it published...

By the way I just figured out a few days ago how it works when publishing an article. You send it in to a given journal, this journal then sends it to a researcher acive in that field (usually a professor I guess), who then gives feed-back directly to you.

 

Is this your personal experience to receive feedback on a submission Sanctus? I ask because it is not my experience or understanding that a submission is edited or reviewed by the journal folks. My experience is that when a submission is rejected that's it; fini. Plain ol' 'we have no use of it' or some such. :shrug:

 

Craig's plan sounds pretty good for what we can do here for you Don, short of you giving out your work. As to the work having uses in cryptography, that's a rather sticky wicket. http://hypography.com/forums/general-science-news/12490-math-plus-cryptography-equals-drama-conflict.html

 

Let the countdown begin! :eek2: :(

[CraigD]

I wouldn’t worry about the computer security implication of an algorithm to generate consecutive prime numbers.

 

The major thread to most popular public-key encryption systems, such as RSA, is the invention of a much more efficient solution to the factoring problem. The output of a prime number generator, regardless of how efficient it is, doesn’t solve the factoring problem. Since, per the prime number theorem the number of prime numbers from 2 to x is known to be near [math]\frac1{\ln x}[/math], for a prime number generator to be any better for brute-force factoring of a typical cryptographic composite number (eg: for RSA-4096) than counting by odd numbers, it must require no more that about 710 ([math]\ln 2^{1024}[/math]) times the computing effort incrementing by 2. Even if a prime number generator were this efficient, the brute-force factoring approach with which it would help would only be slightly helped. For common cryptographic composite number sizes, such approaches are impractical for any classical (ie: not a quantum computer) computer.

 

Only if a prime number generator also implements or suggests a dramatically more efficient solution to the factoring problem is there a need to worry about its computer security implications.

[C1ay]

Don,

 

You could always use your algorithm to simply generate the next record twin primes, Fermat prime, Mersenne prime or some other that currently has a reward offered for it. You'd get money and fame in short order.

[Don Blazys]

To:Buffy,

 

While I don't know all of the consequences of my formula, I do know some of them.

 

For instance, I know that there is an extraordinarily surprising connection between the entire sequence of primes and the numbers Pi and e.

 

I also know that it shares a very important property with the logarithmic integral Li(x), which would make it "of interest" to mathematicians who are studying or working on related problems such as the Riemann Hypothesis.

 

Good point about the usefullness of my formula with regards as to how far todays computers can use it to generate the sequence of primes.

 

In order to estimate that "usefullness", I first had to do a little "Google searching" to find these facts.

 

(1) The number Pi has been calculated to approximately 10^12 decimal places.

 

(2) The number e has been calculated to approximately 10^11 decimal places.

 

(3) The number of known primes is approximately 2(10^21).

 

Now, my biggest problem is that until very recently, I never even touched a computer, so I don't know much about about them, or their "number crunching" capabilities. However, since my formula involves both Pi and e, it is clear that the number of primes that my formula will generate in sequence is clearly limited to about 10^11.

 

Thus, I need not worry about it's being used to compromise encryption codes.

 

Moreover, even relatively "simple" calculations that involve sines and natural logarithms (as my formula does) can become extraordinarily cumbersome if accuracy to more than a dozen or so decimal places is required. For instance, using the best and fastest computer available to you... to how many decimal places can you or any of the "math denizens" here at Hypography calculate the root of:

 

sin(x^(1/2))-ln(ln(x)) which is approximately x= 6.2207156287787... .

 

As you can see, the best that I can do with my TI-89 is only 13 decimal places, and I can't even be sure that the last several digits are correct!

 

I would definitely be interested in exploring my formula with anyone who can do significantly better than my measly 13 digit calculation.

 

So there it is. Your most exellent point has resulted in the conclusion that I can definitely publish my formula in this and in other forums because at present, it can't possibly be used for nefarious purposes and security companies will not be interested in buying it.

 

I'm sure that my wife will be disappoined, but after three months, the mathematicians at Princeton have yet to point out a flaw in my proof of the BC, and there are now four more very good mathematicians who have verified it, so she and I can still hope for a somewhat smaller, yet substantial windfall.

 

Thanks for making, what turned out to be, a most enlightening and helpfull point. I will talk about how to present my discovery to the public with my wife, and a friend of mine who happens to be a retired judge.

 

I really want every math lover to see it soon, maybe real soon!

 

Don.

[sanctus]

Turtle, it is the experience of the people I'm working with (I am still working on my first article and since I am doing a PhD at a University people I work with know what they are talking about). If it is rejected, then may be you don't get feed-back, but it can also be accepted with the comments by the referee that you have to change that, that and that...

[Don Blazys]

To: Sanctus, C1ay, CraigD, Turtle, Pyrotex, Qfwfq, Essay and of course Buffy,

 

Thanks for all the advice and input. You were all very kind, and I am very gratefull. As of now, I really don't see any reason why I can't simply self publish my formula here and in other forums.

 

The journals can wait!

 

I personally don't know anyone who reads those stuffy journals anyway, but if Turtle or another math denizen would like to help me do all the stuff that I can't, such as putting it in LaTex, then I would be most gratefull.

 

At this point I'm almost certain that I will publish it here and in other forums in about a week. If anyone knows of a reason why I shouldn't, then please let me know before then.

 

Meanwhile, can anyone beat my measly 13 decimal place calculation in my previous post?

 

Don.

[Turtle]

Turtle, it is the experience of the people I'm working with (I am still working on my first article and since I am doing a PhD at a University people I work with know what they are talking about). If it is rejected, then may be you don't get feed-back, but it can also be accepted with the comments by the referee that you have to change that, that and that...

 

Well, that's my point. Getting something published in a professional journal is really not about the veracity of the math (or whatever subject), it's about the politics & money of publishing. The journals are in it for business and they have only so much space each edition and so those on the inside get an edge on that space. Then there is who reads/subscribes to these journals and again -save for someone going to a library- it is the insiders who read them.

 

In regard to Don's discovery and copyright, everyone owns the copyright to whatever they produce, but the problem is protecting it which means if someone violates it you have to hire a lawyer and sue. Not much different for a patent either except one has to first hire a patent attorney to search & see if it's already been done, file the patent , pay more, wait, wait, and then wait some more until someone steals the material and then you goo with the copyright deal and hire an attorney & sue.

 

In watching a show recently on fractals & Mandelbrot, they said the journal editors repeatedly refused his work and castigated & mocked him for his trouble ta boot. He wrote his own book & found a publisher outside academia.

 

In conclusion, the internet has made the world flat and the journals obsolete.

[Qfwfq]

For instance, using the best and fastest computer available to you... to how many decimal places can you or any of the "math denizens" here at Hypography calculate the root of:

 

sin(x^(1/2))-ln(ln(x))= 6.2207156287787... ?

 

As you can see, the best that I can do with my TI-89 is only 13 decimal places, and I can't even be sure that the last several digits are correct!

Well, despite your misleading equality, I supposed and verified that you mean finding a zero of the expression:

 

[math]\sin x^{\frac12}-\ln\ln x[/math]

 

which can be done by methods such as those of Newton or Gauss; in some cases even regula falsi or the secant method may suffice. The standard computational type double has a 53 bit mantissa, it can improve precision somewhat compared with your result. If you really find it important to improve precision on that computation you could always contruct an ad hoc numeric type or use a language which handles higher precision, perhaps Craig's favourite language would suit the purpose.