The Holy Grail Of Mathematics.

[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 , 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 () 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]

Originally Posted by 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...

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]

Originally Posted by Don Blazys

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:



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.

[Don Blazys]

To:Qfwfq,

Can you show me the most accurate value of that root that you were able to come up with?

Don?

[freeztar]

Originally Posted by Don Blazys

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

If you're using Windows, the built in calculator can do up to 30 decimal places.

[Don Blazys]

To: Freeztar,

Thanks for the information. It's important that at least one mathematician/computer whizz here at hypography be able to determine the root of the expression:

sin(x^(1/2))-ln(ln(x))

which is approximately x= 6.2207156287788...

to as many decimal places as possible, because a similar calculation is necessary in my formula. The better the approximation, the more primes will be generated in order of magnitude! It's really amazing!

Don.