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## Re: [PrimeNumbers] What if Mr. X would have a formula for the prime series?

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• ... There is no such computationally efficient *formula*. However there are _algorithms_, with obvious drawbacks. It s not what most mathematicians want, hope
Message 1 of 16 , Apr 2 12:43 PM
Le 2013-04-02 21:34, viva8698 a écrit :
> Recently we had a great discussion during a meeting with colleague
> matematicians and we opened a theme that seems to get quite an
> interesting intellectual quiz:
>
> What would be the consequences if Mr. X would have a formula to
> calculate from a given prime the whole ordered series of the
> subsequent primes one after the other?

There is no such computationally efficient *formula*.

However there are _algorithms_, with obvious drawbacks.
It's not what most mathematicians want, hope or dream of,
but Mathematics is not created to please us either.
• There are computationally inefficient formulas: Define a function F(a,b) to be equal to a/gcd(a,b^10). Next_prime(x) is then equal to: sum for n=x to Infinity
Message 2 of 16 , Apr 2 4:54 PM
There are computationally inefficient formulas:

Define a function F(a,b) to be equal to a/gcd(a,b^10).

Next_prime(x) is then equal to:

sum for n=x to Infinity of floor(1/F(n!,x!))

I believe this function is correct for all integers x >= 0.

In practice, you only need to sum until you get a zero value,
and all subsequent values will be zero.

What makes the above computationally infeasible, even for relatively
small numbers, is that x! is already beyond representation in any
known computer long before finding primes becomes difficult.

On 4/2/2013 12:43 PM, whygee@... wrote:
> Le 2013-04-02 21:34, viva8698 a écrit :
>> Recently we had a great discussion during a meeting with colleague
>> matematicians and we opened a theme that seems to get quite an
>> interesting intellectual quiz:
>>
>> What would be the consequences if Mr. X would have a formula to
>> calculate from a given prime the whole ordered series of the
>> subsequent primes one after the other?
>
> There is no such computationally efficient *formula*.
>
> However there are _algorithms_, with obvious drawbacks.
> It's not what most mathematicians want, hope or dream of,
> but Mathematics is not created to please us either.
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://primes.utm.edu/
>
> Yahoo! Groups Links
>
>
>
>
>
>
• That formula should read: x + (sum for n=x to Infinity of floor(1/F(n!,x!)))
Message 3 of 16 , Apr 2 5:03 PM
That formula should read:

x + (sum for n=x to Infinity of floor(1/F(n!,x!)))

On 4/2/2013 4:54 PM, Jack Brennen wrote:
> There are computationally inefficient formulas:
>
> Define a function F(a,b) to be equal to a/gcd(a,b^10).
>
> Next_prime(x) is then equal to:
>
> sum for n=x to Infinity of floor(1/F(n!,x!))
>
> I believe this function is correct for all integers x >= 0.
>
> In practice, you only need to sum until you get a zero value,
> and all subsequent values will be zero.
>
> What makes the above computationally infeasible, even for relatively
> small numbers, is that x! is already beyond representation in any
> known computer long before finding primes becomes difficult.
>
>
>
>
> On 4/2/2013 12:43 PM, whygee@... wrote:
>> Le 2013-04-02 21:34, viva8698 a écrit :
>>> Recently we had a great discussion during a meeting with colleague
>>> matematicians and we opened a theme that seems to get quite an
>>> interesting intellectual quiz:
>>>
>>> What would be the consequences if Mr. X would have a formula to
>>> calculate from a given prime the whole ordered series of the
>>> subsequent primes one after the other?
>>
>> There is no such computationally efficient *formula*.
>>
>> However there are _algorithms_, with obvious drawbacks.
>> It's not what most mathematicians want, hope or dream of,
>> but Mathematics is not created to please us either.
>>
>>
>> ------------------------------------
>>
>> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
>> The Prime Pages : http://primes.utm.edu/
>>
>> Yahoo! Groups Links
>>
>>
>>
>>
>>
>>
>
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://primes.utm.edu/
>
> Yahoo! Groups Links
>
>
>
>
>
• Hi, ... yes, however this does not help the original poster in any way because this formula is the mathematical equivalent of a tautology. It does not say what
Message 4 of 16 , Apr 2 5:09 PM
Hi,

Le 2013-04-03 02:03, Jack Brennen a écrit :
> That formula should read:
>
> x + (sum for n=x to Infinity of floor(1/F(n!,x!)))

yes, however this does not help the original poster in any way
because this formula is the mathematical equivalent of a tautology.
It does not say what primes numbers are or are not, does not
explains or brings insight into the puzzling structure of the
intervals/gaps.
At least, a sieve is useful in expressing what primes are not
(they are not composite numbers) :-)

I imagine that the original idea for this thread is that
the magic formula would somehow explain the prime numbers,
revealings new insights, which would make Riemann's Zeta moot,
or something like that.

But these are speculations, of course :-)

yg
• it seems to me that we have often the question is that a formula or an algorithm. Lets explain like this similar we have a rule/formula to find one after the
Message 5 of 16 , Apr 2 10:20 PM
it seems to me that we have often the question is that a formula or an algorithm. Lets explain like this similar we have a rule/formula to find one after the next simply the next Fibanoacci number, Mr. X has a rule to find the next prime number.
He would have indeed a mathematical founcation and a theorem.
Of course that could be turned into an algorithm later but it is clearly to distinguish, what he has.

--- In primenumbers@yahoogroups.com, "viva8698" <vaseghisam@...> wrote:
>
> Recently we had a great discussion during a meeting with colleague matematicians and we opened a theme that seems to get quite an interesting intellectual quiz:
>
> What would be the consequences if Mr. X would have a formula to calculate from a given prime the whole ordered series of the subsequent primes one after the other?
>
> - easily said he gets the 2 and he calcualtes with his formula 3, 5, 7,...
>
>
> The question appeared to us at the first glance as a joke but soon brought us to a very serious discussion and an explosion of ideas and questions, as well. On the top of many question marks a couple of them were quite hot:
>
> 1- Would we need to solve the Riemann hypothesis then at all even if Mr. X and his solution does not provide any way to connect to Zeta?
> 2- Would we need to solve all the mountain of conjectures around the primes? And if yes which one?
> - ...
>
> Indeed one major point of discussion was to remind:
>
> 3- What where Euler, Riemann, and all the many great mathematicians seeking for? And what are they still honestly seeking for, when it comes to primes?
>
> Of course there was some bizar questions as well:
>
> - which price would Mr. X be awarded? - Surely not the Nobel price!
> - how should Mr. X publish his formula? or should he keep it for himself?
>
> I send out this annecdotical theme and beg all of you so much for your scenarios at least for the questions 1, 2 and 3.
>
> I am sure this can get a great discussion.
>
> Look foward.
>
• I received two interesting points raised that let me share: - the primes sequence is not linear, in fact it s anti-linear. My question: But it still could have
Message 6 of 16 , Apr 2 10:54 PM
I received two interesting points raised that let me share:

- the primes sequence is not linear, in fact it's anti-linear.

My question: But it still could have a an iterative rule behind like for instance fractals or another type of topology?

-it's increasing but at a rate that makes it unpredictable linearly.

My question: Where do you see the mathematical proof that it is "unpredictable"?

Lets take the consequences if Mr. X would have the proof it would be a positive answer to both: a clear and precise toplogy and predictable from on to the next.

--- In primenumbers@yahoogroups.com, whygee@... wrote:
>
> Hi,
>
> Le 2013-04-03 02:03, Jack Brennen a Ã©critÂ :
> > That formula should read:
> >
> > x + (sum for n=x to Infinity of floor(1/F(n!,x!)))
>
> yes, however this does not help the original poster in any way
> because this formula is the mathematical equivalent of a tautology.
> It does not say what primes numbers are or are not, does not
> explains or brings insight into the puzzling structure of the
> intervals/gaps.
> At least, a sieve is useful in expressing what primes are not
> (they are not composite numbers) :-)
>
> I imagine that the original idea for this thread is that
> the magic formula would somehow explain the prime numbers,
> revealings new insights, which would make Riemann's Zeta moot,
> or something like that.
>
> But these are speculations, of course :-)
>
> yg
>
• ... It is to be assumed that the formula, which implies an algorithm, is at least as fast as a low degree polynomial to evaluate. ... I would expect that the
Message 7 of 16 , Apr 3 6:52 AM
On 4/3/2013 8:29 AM, primenumbers@yahoogroups.com wrote:
> 1a. What if Mr. X would have a formula for the prime series?
> Posted by: "viva8698"vaseghisam@... viva8698
> Date: Tue Apr 2, 2013 12:34 pm ((PDT))
>
> Recently we had a great discussion during a meeting with colleague matematicians and we opened a theme that seems to get quite an interesting intellectual quiz:
>
> What would be the consequences if Mr. X would have a formula to calculate from a given prime the whole ordered series of the subsequent primes one after the other?
>
> - easily said he gets the 2 and he calcualtes with his formula 3, 5, 7,...

:) This would make a good mathematical fiction story.

It is to be assumed that the formula, which implies an algorithm, is at
least as fast as a low degree polynomial to evaluate.

>
>
> The question appeared to us at the first glance as a joke but soon brought us to a very serious discussion and an explosion of ideas and questions, as well. On the top of many question marks a couple of them were quite hot:
>
> 1- Would we need to solve the Riemann hypothesis then at all even if Mr. X and his solution does not provide any way to connect to Zeta?

I would expect that the formula of Mr. X would necessarily show a
trivial way to prove the Riemann hypothesis.

http://primes.utm.edu/notes/rh.html

> 2- Would we need to solve all the mountain of conjectures around the primes? And if yes which one?

??? Do you mean this question the other way around? Would the formula
of Mr. X, and the associated
theory by which he developed it, easily solve all those other conjectures?

>
>
> Indeed one major point of discussion was to remind:
>
> 3- What where Euler, Riemann, and all the many great mathematicians seeking for? And what are they still honestly seeking for, when it comes to primes?

Perhaps they were looking for that magical formula found by Mr. X.

> Of course there was some bizar questions as well:
>
> - which price would Mr. X be awarded? - Surely not the Nobel price!

Will the associated theory by which Mr. X derived his formula help him
factor arbitrarily large positive integers?

> - how should Mr. X publish his formula? or should he keep it for himself?

It depends on whether or not everyone knowing his formula and associated
theory would allow
crooks to break encryption of electronic commercial transactions.

Kermit Rose
• Wouldn t it be enough if given prime pn the formula calculates pn+1? All tests for primality would have to be reviewed. Any algorithm depending on primality
Message 8 of 16 , Apr 3 10:40 AM
Wouldn' t it be enough if given prime pn the formula calculates pn+1?

All tests for primality would have to be reviewed. Any algorithm depending on primality testing could get a boost in performance from the formula. Cryptography and on line security could be severely affected.

Knowing that no prime is missing in the sequence someone could be tempted to create a new encoding system for numbers in general, based on prime factors: 10 = (1,0,1) 18 = (1,2) 5488 = (4,0,0,3) somehow avoiding the repeating zeros big numbers could be reduced to short sequences.

________________________________
From: viva8698 <vaseghisam@...>
Sent: Tuesday, April 2, 2013 1:34 PM
Subject: [PrimeNumbers] What if Mr. X would have a formula for the prime series?

Recently we had a great discussion during a meeting with colleague matematicians and we opened a theme that seems to get quite an interesting intellectual quiz:

What would be the consequences if Mr. X would have a formula to calculate from a given prime the whole ordered series of the subsequent primes one after the other?

- easily said he gets the 2 and he calcualtes with his formula 3, 5, 7,...

The question appeared to us at the first glance as a joke but soon brought us to a very serious discussion and an explosion of ideas and questions, as well. On the top of many question marks a couple of them were quite hot:

1- Would we need to solve the Riemann hypothesis then at all even if Mr. X and his solution does not provide any way to connect to Zeta?
2- Would we need to solve all the mountain of conjectures around the primes? And if yes which one?
- ...

Indeed one major point of discussion was to remind:

3- What where Euler, Riemann, and all the many great mathematicians seeking for? And what are they still honestly seeking for, when it comes to primes?

Of course there was some bizar questions as well:

- which price would Mr. X be awarded? - Surely not the Nobel price!
- how should Mr. X publish his formula? or should he keep it for himself?

I send out this annecdotical theme and beg all of you so much for your scenarios at least for the questions 1, 2 and 3.

I am sure this can get a great discussion.

Look foward.

[Non-text portions of this message have been removed]
• ... Something like that is used in Fractran (but in reverse) :-) http://en.wikipedia.org/wiki/Fractran yg
Message 9 of 16 , Apr 3 3:29 PM
Le 2013-04-03 19:40, Leonel Morales a écrit :
> Knowing that no prime is missing in the sequence someone could be
> tempted to create a new encoding system for numbers in general, based
> on prime factors: 10 = (1,0,1) 18 = (1,2) 5488 = (4,0,0,3) somehow
> avoiding the repeating zeros big numbers could be reduced to short
> sequences.

Something like that is used in Fractran (but in reverse) :-)

http://en.wikipedia.org/wiki/Fractran

yg
• Dear Kermit, i could not contact you by your old email adress, neither by kermit@polaris.net nor by the prime forum. Please send me your new mail adress to
Message 10 of 16 , Apr 4 12:32 AM
Dear Kermit,

i could not contact you by your old email adress,
neither by kermit@... nor by the prime forum.

pi@...

The best greetings from the primes
Bernhard Helmes
• In reply to What if Mr. X would have a formula for the prime series , please see chapters 2 and 3 of my book Prime Numbers Characteristics -Why They are
Message 11 of 16 , Apr 4 5:20 AM
In reply to "What if Mr. X would have a formula for the prime series", please see chapters 2 and 3 of my book "Prime Numbers' Characteristics -Why They are what they are".
L.J.Balasundaram

[Non-text portions of this message have been removed]
• ... from ... the other? Any of the freely or costly available CAS have this function already implemented, it s called nextprime() (and explicit formulas do
Message 12 of 16 , Apr 4 8:42 AM
> What would be the consequences if Mr. X would have a formula to calculate
from
> a given prime the whole ordered series of the subsequent primes one after
the other?

Any of the freely or costly available CAS have this function already
implemented, it's called nextprime()

(and explicit formulas do exist, although those which remain readable are
much less efficient than those which are less "explicit" but rather written
as algorithm).

I admit that most nextprime() functions use pseudo-primality tests (for
which no counter example is known and which are less likely to yield a
false positive than the probability of an error due to the computer
hardware).
But does this make a difference for this discussion? If so, in which
respect?

> 1- Would we need to solve the Riemann hypothesis

Depends on what you mean by "need".
I don't think that existence of the nextprime() function (or formula)
implies the RH.

> 2- Would we need to solve all the mountain of conjectures around the
primes?

Having a formula which yields all primes one after the other
does not yield a response to many conjectures, I think.

> Indeed one major point of discussion was to remind:
>
> 3- What where Euler, Riemann, and all the many great mathematicians
seeking for?

(Personally I think they were and are seeking better understanding of
several quite distinct mathematical problems, among which might be the
understanding of the structure of the set of prime numbers.
But even knowing all of the latter (even if it was "all at once" and not
"one after the other")
does not mean to understand much about the structure.
That's quite similar with other branches of science:
Even a most complete set of measurements is not equivalent to an
explanation or understanding.)

Regards,
Maximilian

> Look foward.

[Non-text portions of this message have been removed]
• This is absolutely exciting. You bring it to the point of structure What do you see all as the problem of structure We where just having a chat with our
Message 13 of 16 , Apr 5 8:12 AM
This is absolutely exciting.
You bring it to the point of "structure"
What do you see all as the problem of "structure"
We where just having a chat with our colleagues from physics.
They seem to have an interseting annecdote:

Lets assume the Prime Numbers are distributed in a labyrinth with a crayz/chaotic structure and we leave Mr. X there somwhere inside the Labyrinth. He is not allowed at any point (right or left) to make any test/trials (that means e.g. he is not allowed to go first right and proof and get back and go left if first trial wrong). If his formula works he would find the way to all primes (that will lead him to exit). He can only trust his formula (micro).

Would he fail?

The knowledge of the structure would be a probably more powerful (macro), in that case he would have an overall plan (view from top) and know the whole topology. Is that what you meant by structure?

And one of the colleagues came just back with one question: The structure of primes in what?

--- In primenumbers@yahoogroups.com, Maximilian Hasler <maximilian.hasler@...> wrote:
>
> > What would be the consequences if Mr. X would have a formula to calculate
> from
> > a given prime the whole ordered series of the subsequent primes one after
> the other?
>
> Any of the freely or costly available CAS have this function already
> implemented, it's called nextprime()
>
> (and explicit formulas do exist, although those which remain readable are
> much less efficient than those which are less "explicit" but rather written
> as algorithm).
>
> I admit that most nextprime() functions use pseudo-primality tests (for
> which no counter example is known and which are less likely to yield a
> false positive than the probability of an error due to the computer
> hardware).
> But does this make a difference for this discussion? If so, in which
> respect?
>
>
> > 1- Would we need to solve the Riemann hypothesis
>
> Depends on what you mean by "need".
> I don't think that existence of the nextprime() function (or formula)
> implies the RH.
>
> > 2- Would we need to solve all the mountain of conjectures around the
> primes?
>
> Having a formula which yields all primes one after the other
> does not yield a response to many conjectures, I think.
>
>
> > Indeed one major point of discussion was to remind:
> >
> > 3- What where Euler, Riemann, and all the many great mathematicians
> seeking for?
>
> I did not catch what you reminded about this.
>
>
> (Personally I think they were and are seeking better understanding of
> several quite distinct mathematical problems, among which might be the
> understanding of the structure of the set of prime numbers.
> But even knowing all of the latter (even if it was "all at once" and not
> "one after the other")
> does not mean to understand much about the structure.
> That's quite similar with other branches of science:
> Even a most complete set of measurements is not equivalent to an
> explanation or understanding.)
>
> Regards,
> Maximilian
>
>
> > Look foward.
>
>
> [Non-text portions of this message have been removed]
>
• These are exactly such questions which are not at all answered neither by a formula, nor by the (hypothetical and impossible in view of infinitude of primes
Message 14 of 16 , Apr 5 10:00 AM
These are exactly such questions which are not at all answered neither
by a formula, nor by the (hypothetical and impossible in view of
infinitude of primes and finiteness of computers and human mind)
knowledge of ALL primes (which would be way beyond the formula
producing "one prime after the other").

Maximilian

On Fri, Apr 5, 2013 at 11:12 AM, viva8698 <vaseghisam@...> wrote:
> This is absolutely exciting.
> You bring it to the point of "structure"
> What do you see all as the problem of "structure"
> We where just having a chat with our colleagues from physics.
> They seem to have an interseting annecdote:
>
> Lets assume the Prime Numbers are distributed in a labyrinth with a crayz/chaotic structure and we leave Mr. X there somwhere inside the Labyrinth. He is not allowed at any point (right or left) to make any test/trials (that means e.g. he is not allowed to go first right and proof and get back and go left if first trial wrong). If his formula works he would find the way to all primes (that will lead him to exit). He can only trust his formula (micro).
>
> Would he fail?
>
> The knowledge of the structure would be a probably more powerful (macro), in that case he would have an overall plan (view from top) and know the whole topology. Is that what you meant by structure?
>
> And one of the colleagues came just back with one question: The structure of primes in what?
>
>
> --- In primenumbers@yahoogroups.com, Maximilian Hasler <maximilian.hasler@...> wrote:
>>
>> > What would be the consequences if Mr. X would have a formula to calculate
>> from
>> > a given prime the whole ordered series of the subsequent primes one after
>> the other?
>>
>> Any of the freely or costly available CAS have this function already
>> implemented, it's called nextprime()
>>
>> (and explicit formulas do exist, although those which remain readable are
>> much less efficient than those which are less "explicit" but rather written
>> as algorithm).
>>
>> I admit that most nextprime() functions use pseudo-primality tests (for
>> which no counter example is known and which are less likely to yield a
>> false positive than the probability of an error due to the computer
>> hardware).
>> But does this make a difference for this discussion? If so, in which
>> respect?
>>
>>
>> > 1- Would we need to solve the Riemann hypothesis
>>
>> Depends on what you mean by "need".
>> I don't think that existence of the nextprime() function (or formula)
>> implies the RH.
>>
>> > 2- Would we need to solve all the mountain of conjectures around the
>> primes?
>>
>> Having a formula which yields all primes one after the other
>> does not yield a response to many conjectures, I think.
>>
>>
>> > Indeed one major point of discussion was to remind:
>> >
>> > 3- What where Euler, Riemann, and all the many great mathematicians
>> seeking for?
>>
>> I did not catch what you reminded about this.
>>
>>
>> (Personally I think they were and are seeking better understanding of
>> several quite distinct mathematical problems, among which might be the
>> understanding of the structure of the set of prime numbers.
>> But even knowing all of the latter (even if it was "all at once" and not
>> "one after the other")
>> does not mean to understand much about the structure.
>> That's quite similar with other branches of science:
>> Even a most complete set of measurements is not equivalent to an
>> explanation or understanding.)
>>
>> Regards,
>> Maximilian
>>
>>
>> > Look foward.
>>
>>
>> [Non-text portions of this message have been removed]
>>
>
>
• Lets have a vote everybody on one question then: Do we need a new Fundamental Theory (not theorem) of Arithmetics? It looks to me like revolution isrequired
Message 15 of 16 , Apr 5 10:09 AM
Lets have a vote everybody on one question then:

Do we need a new Fundamental Theory (not theorem) of Arithmetics?

It looks to me like revolution isrequired such as we had by Newton bringing accelleration to mechanics, or by Bohr and Schrödinger bringing quantum to mechanics...

Dont we need a discipline that "observes" the system of numbers (N or P) as a scientist would do? I know this is a great tabu to math! But for a moment isnt that we need strong observationout the cage: the view from top on the labyrinth?

--- In primenumbers@yahoogroups.com, Maximilian Hasler <maximilian.hasler@...> wrote:
>
> These are exactly such questions which are not at all answered neither
> by a formula, nor by the (hypothetical and impossible in view of
> infinitude of primes and finiteness of computers and human mind)
> knowledge of ALL primes (which would be way beyond the formula
> producing "one prime after the other").
>
> Maximilian
>
>
> On Fri, Apr 5, 2013 at 11:12 AM, viva8698 <vaseghisam@...> wrote:
> > This is absolutely exciting.
> > You bring it to the point of "structure"
> > What do you see all as the problem of "structure"
> > We where just having a chat with our colleagues from physics.
> > They seem to have an interseting annecdote:
> >
> > Lets assume the Prime Numbers are distributed in a labyrinth with a crayz/chaotic structure and we leave Mr. X there somwhere inside the Labyrinth. He is not allowed at any point (right or left) to make any test/trials (that means e.g. he is not allowed to go first right and proof and get back and go left if first trial wrong). If his formula works he would find the way to all primes (that will lead him to exit). He can only trust his formula (micro).
> >
> > Would he fail?
> >
> > The knowledge of the structure would be a probably more powerful (macro), in that case he would have an overall plan (view from top) and know the whole topology. Is that what you meant by structure?
> >
> > And one of the colleagues came just back with one question: The structure of primes in what?
> >
> >
> > --- In primenumbers@yahoogroups.com, Maximilian Hasler <maximilian.hasler@> wrote:
> >>
> >> > What would be the consequences if Mr. X would have a formula to calculate
> >> from
> >> > a given prime the whole ordered series of the subsequent primes one after
> >> the other?
> >>
> >> Any of the freely or costly available CAS have this function already
> >> implemented, it's called nextprime()
> >>
> >> (and explicit formulas do exist, although those which remain readable are
> >> much less efficient than those which are less "explicit" but rather written
> >> as algorithm).
> >>
> >> I admit that most nextprime() functions use pseudo-primality tests (for
> >> which no counter example is known and which are less likely to yield a
> >> false positive than the probability of an error due to the computer
> >> hardware).
> >> But does this make a difference for this discussion? If so, in which
> >> respect?
> >>
> >>
> >> > 1- Would we need to solve the Riemann hypothesis
> >>
> >> Depends on what you mean by "need".
> >> I don't think that existence of the nextprime() function (or formula)
> >> implies the RH.
> >>
> >> > 2- Would we need to solve all the mountain of conjectures around the
> >> primes?
> >>
> >> Having a formula which yields all primes one after the other
> >> does not yield a response to many conjectures, I think.
> >>
> >>
> >> > Indeed one major point of discussion was to remind:
> >> >
> >> > 3- What where Euler, Riemann, and all the many great mathematicians
> >> seeking for?
> >>
> >> I did not catch what you reminded about this.
> >>
> >>
> >> (Personally I think they were and are seeking better understanding of
> >> several quite distinct mathematical problems, among which might be the
> >> understanding of the structure of the set of prime numbers.
> >> But even knowing all of the latter (even if it was "all at once" and not
> >> "one after the other")
> >> does not mean to understand much about the structure.
> >> That's quite similar with other branches of science:
> >> Even a most complete set of measurements is not equivalent to an
> >> explanation or understanding.)
> >>
> >> Regards,
> >> Maximilian
> >>
> >>
> >> > Look foward.
> >>
> >>
> >> [Non-text portions of this message have been removed]
> >>
> >
> >
>
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