[For clarity I switch to "&&&" to separate articles here -- djr]
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From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Newsgroups: sci.math
Subject: A Tale of Two Conjectures.
Date: 15 Oct 1996 11:19:24 GMT

======================================================================
This article is an elaboration of part of a recent one by Tony Forbes.
======================================================================


                 ============================
                 THE STORY OF TWO CONJECTURES
                 ============================      W.F.C.Taylor


It is still unknown whether or not there are infinitely many twin primes.

That is to say, no PROOF is known; but it IS "known", (as much as almost
anything in maths is), that there ARE infinitely many.  It just hasn't
been proved yet.  And, not only is it certain to be true, but it is
moreover known exactly what the limiting density of twin primes is.

Just as
            #{ p<n | p is prime}     1
        lim -------------------- = ------
       n->oo        n              log(n)

is known, and provable, so is


            #{ p<n | p & p+2 both prime}      c
        lim ---------------------------- = --------
       n->oo            n                  log(n)^2

known (but not proved).  It is known because

(i) the same heuristic that proves the former also applies with equal
    force to the latter, (and gives the constant c, about 1.2), it just
    doesn't apply *quite* well enough to give a formal proof;

(ii) computer simulations verify the result to within the expected
     probabilistic deviation, as far as (increasingly) it is tested.

It is nothing short of completely & maddeningly frustrating, that the
method resists extending to a rigorous proof!  But all who delve into
such things know it to be true.

-----------------------------------------------

Apparently the first to make such a conjecture were Hardy & Littlewood,
who proved so many other milestone results in prime number theory. They
conjectured that all feasible "constellations", such as twins, p & p+2,
admitted not only infinitely many primes, but also with the appropriate
limiting density.

A "feasible" pattern is a finite sequence of integers, that doesn't cover
every possible residue [mod n] for any n>1.      This heavy sounding
condition is actually rather trivial, merely designed to exclude
"impossible prime patterns" such as  p, p+2, p+4,  which can only all be
prime for p=3; otherwise one is always composite, divisible by 3.  Similarly
p, p+2, p+6, p+8, p+14  is impossible, as one must be divisible by 5.

But apart from such cases, any pattern is conjectured to have infinitely
many prime instances.  For example...

p  p+2            TWINS      e.g.  5,7   11,13   17,19  etc

p  p+2  p+6  p+8  QUARTETS   e.g.  11,13,17,19     101,103,107,109  etc

...and so on.  Each feasible constellation has limiting density

            #{ p<n | p+k all prime for k in the pattern}     c_P
        lim -------------------------------------------- = --------
       n->oo                     n                         log(n)^P

when P is the size of the pattern, & c_P a constant depending on the pattern.

So that is Hardy & Littlewood's first conjecture.

------------------------------------------------

Their second conjecture is even simpler.

It seeks to codify the natural intuition, also born out by copious
computer searches, that the primes are "thicker on the ground" in any
stretch near the beginning of N, than any similar sized stretch later on.

Formally:    #{ prime p | 1 < p < n }  >  #{ prime p | k+1 < p < k+n }

             for all  k and n.

As I say, this second conjecture is also highly believable, and supported
by computer searches, and always with a great deal of leeway to spare.

_____________________________________________________________
BUT: since 1989 it has been known that
                   AT LEAST ONE OF THESE CONJECTURES IS FALSE!
-------------------------------------------------------------

In that year, someone managed to root out an enormously large feasible
pattern, such that, if even one instance of it were all prime, (much less
infinitely many!), then the second conjecture must be false.

The numbers involved were enormous; but just last year, a vastly simpler
pattern was discovered.  And it is not yet clear that some even lower
pattern may exist, of a similar type!

The new example is so small it is easily within the range of "Maple" at
1 second/slash; or your own efforts on your home computer for an evening.
Here it is.

The feasible pattern is

{-24049, -24043, ..., -1223, - 1217,   1217, 1223, ..., 24043, 24049};

where this is in fact precisely

{-p_2675, -p_2674,... -p_200, -p_199,   p_199, p_200,... p_2674, p_2675};

where p_n is the n-th prime.  Because these numbers are themselves prime,
it is clear that no number can divide any, hence it is a feasible pattern.
Thus, by H&L's first conjecture, there is a sequence of primes somewhere
that is in this pattern; and it will contain  4954  primes, in an
interval of width  48098.


BUT!    p_4954 = 48109.       So the FIRST 4954 primes are
                              2,3,5,7,11, ... ,48109;  and these occur in an
interval of width  48107 (!)
This is decidely wider, and with a bit to spare.

So the second H&L conjecture would be FALSE!    Remarkable.

------------------------------------------------------------

Poor old Hardy & Littlewood!  But in a way it is fitting - they will
gain much more fame for these incompatible conjectures, than those boring
types who make such tediously trivial conjectures that they always turn
out to be true.

But which (if either!) IS true?

All the smart money is on the first conjecture being true.  It is the one
with both (1) a theoretical justification, & (2) better search accuracy.
The second conjecture is just another case of a false conjecture being
suggested by the now famous "law of small numbers", (a form of Sod's law).



One last lingering question is... where is the first *actual* set of primes
in this pattern, that specifically fails the second conjecture?

Alas!  Don't expect to find one all that soon!   The density results above
suggest that the first one should start somewhere about...
      25000
p = 10     ; & to find 4954 consecutive primes in *just* the right pattern,
             way up there, is NOT on the Cray agenda for this century...

=============================================================================

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From: gerry@mpce.mq.edu.au (Gerry Myerson)
Newsgroups: sci.math
Subject: Re: A Tale of Two Conjectures.
Date: 16 Oct 1996 01:05:33 GMT

In article <53vrvs$nf6@cantuc.canterbury.ac.nz>,
mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
> 
[description of two plausible conjectures about prime numbers, deleted]

> BUT: since 1989 it has been known that
>                    AT LEAST ONE OF THESE CONJECTURES IS FALSE!

1972, actually. Here are some references: 

Douglas Hensley & Ian Richards, On the incompatability of two conjectures 
concerning primes, Analytic Number Theory (Proc. Sympos. Pure Math., Vol. 
XXIV, St. Louis Univ., 1972), 123--127. Amer. Math. Soc., 1973. 
MR 49 #4950. 

Douglas Hensley & Ian Richards, Primes in intervals, Acta Arith. 25 
(1973/74) 375--391. MR 53 #305. 

Ian Richards, On the incompatability of two conjectures concerning primes; 
a discussion of the use of computers in attacking a theoretical problem, 
Bull. Amer. Math. Soc. 80 (1974) 419--438. MR 49 #2601. 

> The numbers involved were enormous; but just last year, a vastly simpler
> pattern was discovered.  

References?

> All the smart money is on the first conjecture being true.  

Around 1979 a philosophy-of-math type gave a talk at SUNY Buffalo about 
conjectures in math, and after the talk I told him about the Hensley-
Richards result. His immediate reaction was that of course the second 
conjecture was the true one, and he was quite surprised when I told him 
that he was contradicting the prevailing view among the experts. 

Gerry Myerson (gerry@mpce.mq.edu.au)
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From: cet1@cus.cam.ac.uk (Chris Thompson)
Newsgroups: sci.math
Subject: Re: A Tale of Two Conjectures.
Date: 16 Oct 1996 15:02:36 GMT

In article <53vrvs$nf6@cantuc.canterbury.ac.nz>, mathwft@math.canterbury.ac.nz 
(Bill Taylor) writes:
[...] 
|>           
|> That is to say, no PROOF is known; but it IS "known", (as much as almost
|> anything in maths is), that there ARE infinitely many.  It just hasn't
|> been proved yet.  

[and much in similar vein]. I have to say I think this use of language
is very unhelpful. The infinitude of prime pairs is a "very probable
conjecture", "morally certain", or what have you, but the one thing it
isn't is "known". 

Except by the Supreme Fascist, that is, and I sort of got the impression,
Bill, that you weren't on speaking terms with her?

|> The feasible pattern is
|> 
|> {-24049, -24043, ..., -1223, - 1217,   1217, 1223, ..., 24043, 24049};
|> 
|> where this is in fact precisely
|> 
|> {-p_2675, -p_2674,... -p_200, -p_199,   p_199, p_200,... p_2674, p_2675};
|> 
|> where p_n is the n-th prime.  Because these numbers are themselves prime,
|> it is clear that no number can divide any, hence it is a feasible pattern.

Hang on a moment there. Feasibility means that for any prime p, the 4954
numbers do not exhaust all residue classes mod p. This is "clear" for
p <= p_198 (because 0 mod p is not covered) and for p > 4954, but for
the primes in between it definitely has to be checked.

In article <gerry-161096105323@e7a4mac05.mpce.mq.edu.au>, gerry@mpce.mq.edu.au 
(Gerry Myerson) asks for

#> References?

In e-mail which Tony Forbes sent to me and Bill Taylor [I don't think this
was ever a Usenet article] he mentions

+> [...] the following result (published in M500, issue 146,
+> September 1995)

"M500" is apparently a serial associated with the Open University.

Chris Thompson
Email: cet1@cam.ac.uk

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From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Newsgroups: sci.math
Subject: Re: A Tale of Two Conjectures.
Date: 25 Oct 1996 01:10:56 GMT

gerry@mpce.mq.edu.au (Gerry Myerson) writes:

|> > BUT: since 1989 it has been known that
|> >                    AT LEAST ONE OF THESE CONJECTURES IS FALSE!
|> 
|> 1972, actually. Here are some references: 
|> 
|> Douglas Hensley & Ian Richards, Primes in intervals, Acta Arith. 25 
|> (1973/74) 375--391. MR 53 #305. 

Yes, thanks for that.  Soon after I posted I remembered it appears in Guy's
Unsolved Problems in Num Theory, with refs.     Yes, 1972; boo-boo by me!


|> > The numbers involved were enormous; but just last year, a vastly simpler
|> > pattern was discovered.  
|> 
|> References?

Tony Forbes (tonyforbes@ltkz.demon.co.uk) who wrote the original article here,
appears to be the discoverer; he gave M500, issue 146, September 1995, being...

>The M500 Society is a mathematical society for students, staff and
>friends of the Open University. "M500" is its newsletter.

-----------

|> > All the smart money is on the first conjecture being true.  
|> 
|> Around 1979 a philosophy-of-math type gave a talk at SUNY Buffalo about 
|> conjectures in math, and after the talk I told him about the Hensley-
|> Richards result. His immediate reaction was that of course the second 
|> conjecture was the true one, and he was quite surprised when I told him 
|> that he was contradicting the prevailing view among the experts. 

How typical!  Reinforcing my impression that *only* professional mathematicians
should do philosophy of math.   Same probably goes for *any* "philosophy-of-"
subject.  What's the good of waffling on about something if you don't
really know what you're talking about?


Nothing like a good rant...

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            Bill Taylor               W.Taylor@math.canterbury.ac.nz
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