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Two Hardy-Littlewood Conjectures
From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: A Tale of Two Conjectures.
Newsgroups: sci.math
Date: 15 Oct 1996 11:19:24 GMT
Organization: Department of Mathematics and Statistics,
University of Canterbury, Christchurch, NZ.
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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
---------------------- ~ ------
n log(n)
is known, and provable, so is
#{ p < n | p and p+2 both prime} c
-------------------------------- ~ --------
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 and 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 and Littlewood,
who proved so many other milestone results in prime number theory. They
conjectured that all feasible "constellations", such as twins, p and 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
--------------------------------------------- ~ --------
n log(n)^P
when P is the size of the pattern, and c_P a constant depending on the
pattern.
So that is Hardy and 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 Hardy and Littlewood'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 Hardy-Littlewood conjecture would be FALSE! Remarkable.
------------------------------------------------------------
Poor old Hardy and 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, and (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 p = 10^25000;
and to find 4954 consecutive primes in *just* the right pattern, way up
there, is NOT on the Cray agenda for this century...
=============================================================================
From: gerry@mpce.mq.edu.au (Gerry Myerson)
Subject: Re: A Tale of Two Conjectures.
Newsgroups: sci.math
Date: 16 Oct 1996 01:05:33 GMT
Organization: CeNTRe for Number Theory Research
In article <53vrvs$nf6@cantuc.canterbury.ac.nz>,
mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
>
[description of two plausible conjectures about prime numbers, deleted]
> BUT: since 1989 it has been known that
> AT LEAST ONE OF THESE CONJECTURES IS FALSE!
1972, actually. Here are some references:
Douglas Hensley and Ian Richards, On the incompatability of two conjectures
concerning primes, Analytic Number Theory (Proc. Sympos. Pure Math., Vol.
XXIV, St. Louis Univ., 1972), 123--127. Amer. Math. Soc., 1973.
MR 49 #4950.
Douglas Hensley and Ian Richards, Primes in intervals, Acta Arith. 25
(1973/74) 375--391. MR 53 #305.
Ian Richards, On the incompatability of two conjectures concerning primes;
a discussion of the use of computers in attacking a theoretical problem,
Bull. Amer. Math. Soc. 80 (1974) 419--438. MR 49 #2601.
> The numbers involved were enormous; but just last year, a vastly simpler
> pattern was discovered.
References?
> All the smart money is on the first conjecture being true.
Around 1979 a philosophy-of-math type gave a talk at SUNY Buffalo about
conjectures in math, and after the talk I told him about the Hensley-
Richards result. His immediate reaction was that of course the second
conjecture was the true one, and he was quite surprised when I told him
that he was contradicting the prevailing view among the experts.
Gerry Myerson (gerry@mpce.mq.edu.au)
=============================================================================
From: cet1@cus.cam.ac.uk (Chris Thompson)
Subject: Re: A Tale of Two Conjectures.
Newsgroups: sci.math
Date: 16 Oct 1996 15:02:36 GMT
Organization: University of Cambridge, England
In article <53vrvs$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".
...
|> 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
Copyright © 1995-2001 Steven Finch
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