HDTenEightyP/NetNews huggingface dataset:
From: frank@Cookie.secapl.com (Frank Adams)
Subject: Re: Fundamental Theorem of Geometry (and others)
Date: Mon, 17 Aug 1992 18:43:28 GMT
Organization: Security APL, Inc.
Keywords: fundamental theorem

In article <1992Aug5.001654.27795@informix.com> proberts@informix.com (Paul Roberts) writes:
>In article <1992Aug4.183519.2276@u.washington.edu> petry@frobenius.math.washington.edu (David Petry) writes:
>>In article <1992Aug4.135650.147@csc.canterbury.ac.nz> wft@math.canterbury.ac.nz (Bill Taylor) writes:
>>>             ..... and for the big daddy of them all,  what is the 
>>>
>>>       FUNDAMENTAL THEOREM OF MATHEMATICS ??
>>>       ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>>
>>My vote goes to the principle of induction.
>
>But is it a theorem; can it be proved? Or is it not rather an
>axiom about, you know, the integers ....

It all depends.  In pure number theory, it's an axiom.  In set theory, it's
a theorem.

Really, the difference between theorems and axioms is not so fundamental.
If you take a different set of axioms for, say, fields , with the same
consequences, you still have the same theory.  Even a slightly different
theory is not really significant: if you skip the axiom 0 != 1 in field
theory, you have the theory of "fields plus the trivial ring".  A few
theorems and definitions become slightly easier to state, and a few become
slightly harder; but that's all.

"Foundations" are embellishments, not the essence.

(BTW, if there is a "Fundamental Theorem of Mathematics", I think induction
has got to be it.)
Newsgroups: sci.math
From: roach@bikini.wri.com (Kelly Roach)
Subject: Re: u(v^n)w prime puzzle
Organization: Wolfram Research, Inc.
Date: Wed, 19 Aug 1992 22:19:46 GMT

In article <1992Aug19.141957.17410@ecrc.de> jeanmarc@ecrc.de (Jean-Marc  
Andreoli) writes:
> Let x_n be the number u(v^n)w.
> hence x_{n+1} - 10^r x_n = (vw - wz)
> More generally, we have x_{n+1} = a x_n + b
> and we have to prove that such a sequence is either constant or
> contains at least one composite, for any integral values of
> x_0,a,b (with a>1).
> ...
> QED


     Looks like you've got it.  Not the proof I had in mind,
but still, I think it is correct.  Perhaps I'll still post my
solution Friday so that others understand what I was hinting
at.



> Now, I propose the following question:
> 
> Is it true that any sequence x_{n+1} = a x_n + b is either
>constant or contains *infinitely many* composite ? *almost
>only* composite (i.e. it contains only a finite number of primes) ?


      I think I can answer the first question quickly.
I will just dismiss the cases a=-1,0,1 to the readers.
Certainly, x_0=1 with x_{n+1}=-x_n is pretty boring.
Also, we may suppose (a,b)=1 since (a,b)!=1 is not
interesting.  My restriction |a|>1 makes |x_n|<=1 for
all n impossible.  If necessary drop some of the first
few terms of {x_i} and renumber the sequence so that we
have |x_0|>1 and also (x_0,a)=1.  The latter relation is
trivial to achieve because (a,b)=1.  If you doubt I
can make |x_0|>1, note |a|>1 and see (2) below.
     Since |x_0|>1 and also (x_0,a)=1, I can pick a
prime p such that

(1)  p | x_0     and      p does not divide a

We see by the difference equation x_{n+1} = a x_n + b
that

(2)  x_n = a^n*x_0 + b*Sum[a^i,{i,0,n-1}]

Now this Sum is a geometric series and by (1)

     a^i (mod p)

is a periodic function in i.  You can read my solution
to the u(v^n)w puzzle on Friday to find out that

(3)  Sum[a^i,{i,0,n-1}] = 0 (mod p)     if p(p-1) | n.

or prove it yourself.  Hence, by (1), (2), and (3), we
have

(4)  p | x_n     if p(p-1) | n

By |a|>1 and (2), it is clear that |x_n| -> infinity,
so it follows by (4) that infinitely many of the x_n
are composite.
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Evaluation of an integral
Date: 20 Aug 1992 10:55:44 -0400
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu
Keywords: exponential


The indefinite integral of exp(exp(x)) is not an elementary function.
It can be written in terms of the "exponential integral" function Ei.
Here is what Maple told me:

> int(exp(exp(x)),x=0..1/2);
                       Ei(exp(1/2)) - Ei(1)

_
> evalf(");
                           1.862435425

_
> evalf("",500);
1.86243542410770481303208319610528784480767978806791677492786511\
6756585929911929584000958223836189072911105728790853406648259328\
5160931747066377341039540685553425675519341357883987318612260687\
7240917117028993932573426557682047900448981801741402940554396575\
3757565242095688007336161253448510962393332822684592137177470401\
7628981687136387893692977513705639582896999666998738745059455124\
2820516352293270980363589870785304954163909314696773205898299141\
48192082991352267983908311939018676600101681264222032

_
>
-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Newsgroups: sci.math
From: wall@src.honeywell.com (Joe Wall III)
Subject: Univ. of Cambridge Question
Organization: Honeywell Systems & Research Center
Date: Wed, 12 Aug 1992 21:10:11 GMT

If any kind soul is familiar with the Mathematical Tripos at
the University of Cambridge and has time to answer a few questions,
email me a note.

Mathematics professors at Cambridge are encouraged to apply. ;-)

Thanks,

Joe Wall

wall@src.honeywell.com
From: tarokh@server.uwindsor.ca (TAROKH VAHID             )
Newsgroups: sci.math
Subject: Infinite dimensional simple representations of solvable Lie algebras
Keywords: Searching for literature
Date: 12 Aug 1992 17:47:52 GMT

Dear Readers,

Could anybody please tell me what is known about simple representation of
solvable Lie algebras (infinite dimensional case)? Is there a realization
of these or not? If there is what literature it could be found in?

Thank You.

Vahid Tarokh
From: brihut@pro-palmtree.socal.com (Brian Hutchings)
Newsgroups: sci.math
Subject: Morley's theorem; every body knows it -- correct?
Date: Mon, 17 Aug 92 03:19:04 PDT
Organization: The Palmtree BBS - 310/453-8726 - Santa Monica, CA

I've been rackin`m`brain to visualize a spatial context
for the flatland version o`this; any suggestion?...  in the case that
not every last one o`you knows it, it states (what I give
without proof; I really don't recall it, at all) that
the trisectors of the angles of any triangle (adjacently) meet
to form the vertices of a regular (or equi-angular) triangle.

the prize is that of better-knowing the mind of Allah (or
what ever y`call It .-)
Newsgroups: sci.math
From: cc869@cleveland.Freenet.Edu (Udo Sprute)
Subject: Re: Ellipsoid formula wanted
Organization: Case Western Reserve University, Cleveland, Ohio, (USA)
Date: Wed, 12 Aug 92 15:18:00 GMT


referrences:
B.C. Carlson Duke Mathematical Journal V31 (1964) p 405-419
B.C. Carlson Proc. American Mathematical Society V17 (1966) p.32-39
D.H. Lehmer Canadian Jour. Math. V2 (1950) p.267-282
B.C. Carlson Journal of Math. and Physics V44 (1965) p36 -
B.C. Carlson  Special functions and ... 1977, chapter 9 (a book 
                 with the above ref. and further ones)
I think now you got enough  :-)
Special thanks to Torsten Sillke for his literature search!
From: kibler@turing.ICS.UCI.EDU (Dennis Kibler)
Subject: Primes
Newsgroups: sci.math
Date: 17 Aug 92 17:36:15 GMT

In mathematica the number of primes less than n or equal to n
is given by PrimePi[n].

The following took about 1 second to compute.



 Do[Print[i," ",PrimePi[2^i]],{i,30}]
1 1
2 2
3 4
4 6
5 11
6 18
7 31
8 54
9 97
10 172
11 309
12 564
13 1028
14 1900
15 3512
16 6542
17 12251
18 23000
19 43390
20 82025
21 155611
22 295947
23 564163
24 1077871
25 2063689
26 3957809
27 7603553
28 14630843
29 28192750
30 54400028

Newsgroups: sci.math
From: buchino@magnus.acs.ohio-state.edu (Bert N Uchino)
Subject: Re: mathematician's marrying ages.
Organization: The Ohio State University
Date: Thu, 20 Aug 1992 07:02:11 GMT

As many researchers know, prospective studies are very expensive to conduct.   
One potential statistical technique for evaluation directional relationship is 
covariance structural modeling (CSM).  Although I know of no study that has
used CSM on large n cross-sectional data (if anyone knows of such studies 
please let me know), such studies may shed more light on
the issue of "causality" between marital status and mortality/health.  I do
think, however, that their are important theoretical reasons for suggesting a
causal relationship.  For example, the data on social support and health are
relevant, as well as research on stress because some have argued that
loneliless in itself is a stressor with deleterious effects.  Overall, I am
impressed with the consistency in the body of research on marital status and
health.
Newsgroups: sci.math
From: dak@tabaqui.informatik.rwth-aachen.de (David Kastrup)
Subject: Re: Help - non-integral power of a matrix?
Keywords: matrix
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen
Date: 13 Aug 92 06:53:13 GMT

Well, suppose that you have an Eigenvalue decomposition
A = TDT^{-1}, where T is the matrix of Eigenvectors, D is a strictly diagonal
matrix with the Eigenvalues. Works only if your MAtrix is decomposable
in such a way, that is, HAS N different Eigenvectors if its size is NxN.

Then A^a would simply be TD^aT^{-1}, where powering a diagonal matrix is
done by powering its diagonal elements.
Newsgroups: sci.math
From: tordm@vana (Tord Malmgren)
Subject: Re: Did Gauss ever say ...
Reply-To: TordM@VanD.PhySto.SE
Organization: Department of Physics, University of Stockholm -- Sweden
Date: Mon, 17 Aug 1992 20:36:32 GMT

In article <1992Aug17.172858.3795@hubcap.clemson.edu>, steve@hubcap.clemson.edu ("Steve" Stevenson) writes:

>Bartlett's Familar Quotations give Gauss credit for the statement
>"Mathematics is the queen of the sciences."
>I heard anecdotally that he also said "And number theory is the queen
>of mathematics." 

 not a reference, but I heard he made the statement "Arithmetics is the Queen
of Mathematics"....





---------------+--------------------------------
 Tord Malmgren | InterNet: TordM@VanD.PhySto.SE | These opinions are my own,
               | BITNet  : TordM@SESUF51        | and NOT of this department!
---------------+--------------------------------
 Department of Physics, University of Stockholm
From: dorsey@grapevine.EBay.Sun.COM (Bill Dorsey)
Newsgroups: sci.math
Subject: period of a modulo function
Date: 20 Aug 92 08:54:41 GMT
Organization: Sun Microsystems Federal, Milpitas, CA.
NNTP-Posting-Host: grapevine.ebay.sun.com

I have:

	f(x) = a % x^n

where % is the modulo function, a is a prime number >> x, and n is a small
postive integer.  For certain values of n, I observe an obvious periodicity
of f(x).  Is there an equation that describes the period of such an f(x)?
If so, what is it, or kindly refer me to a text that would contain such
information or the means to derive it.

Thanks.

- Bill Dorsey   (dorsey@ebay.sun.com)
Newsgroups: sci.math
From: cgodsil@watserv1.uwaterloo.ca (C. Godsil - C and O)
Subject: Re: A Word-Problem
Organization: University of Waterloo
Date: Wed, 12 Aug 1992 20:27:15 GMT

In article <1992Aug9.172411.26212@unibi.uni-bielefeld.de> uphya159@unibi.uni-bielefeld.de (0118) writes:
>A Word-Problem:
>
>Let  G = < a,b | aabb=baba, bbaa=abab >  the free group of
>two generators a and b with the two relations aabb=baba and bbaa=abab.
>
>Show that  aaabbb unequal bababa.
>
>I've tried all homomorphisms from G to S10 without success.
>(e.g.  a->(123), b->(142)  shows  ab unequal ba  but  aaabbb=bababa)
>
>Torsten Sillke
>(posted by Udo Sprute,  uphya159@unibi.HRZ.Uni-Bielefeld.DE)

The group  G = < a, b : a^2*b^2=(b*a)^2, b^2*a^2=(a*b)^2 >  is soluble:

Let  K  be the subgroup generated by  p = a*b  and  q = b*a .
This subgroup is normal,  with quotient  G/K  cyclic
(NB:  p^a = q,  p^b = p^-1*q^2,  q^a = q^-1*p^2,  &  q^b = p ).

Let  L  be the subgroup generated by  u = p^3  and  v = q^3 .
This subgroup too is normal (with  u^a = v = u^b  and  v^a = u = v^b ),
and also Abelian (since in fact both p and q centralize both u and v).

Further, the quotient  K/L  is a homomorphic image of the (3,3,3) triangle
group  < p, q : p^3 = q^3 = (p*q)^3 = 1 >,  which is Abelian-by-cyclic,
and is therefore soluble.

Now since  G/K is cyclic,  K/L is soluble,  and  L is Abelian, 
it follows that  G itself is soluble.


Here's a finite matrix representation of  G  over GF(2):

  A = [ 0  1  0  0  0 ]            B = [ 1  0  1  1  0 ]
      [ 0  0  1  0  0 ]                [ 0  0  1  0  0 ]
      [ 1  0  0  0  0 ]                [ 1  0  0  0  0 ]
      [ 0  0  0  1  0 ]                [ 1  1  1  0  0 ]
      [ 0  0  0  0  1 ]                [ 0  0  0  0  1 ]

  A^3*B^3 = [ 0  1  1  1  0 ]      B^3*A^3 = [ 0  1  1  1  0 ]
            [ 1  0  1  1  0 ]                [ 1  0  1  1  0 ]
            [ 1  1  0  1  0 ]                [ 1  1  0  1  0 ]
            [ 1  1  1  0  0 ]                [ 1  1  1  0  0 ]
            [ 0  0  0  1  1 ]                [ 1  1  1  0  1 ]

  (A^3*B^3)^2 = (B^3*A^3)^2 = [ 1  0  0  0  0 ]
                              [ 0  1  0  0  0 ]
                              [ 0  0  1  0  0 ]
                              [ 0  0  0  1  0 ] 
                              [ 1  1  1  1  1 ] 
							  
These matrices generate a factor group of order 96, with centre of order 2,
and clearly isomorphic to a subgroup of AGL(4,2). 

Marston Conder  (marston@dibbler.uwaterloo.ca)        12 August 1992
From: pa@verano.sba.ca.us (Pierre Asselin)
Newsgroups: sci.math
Subject: Re: Help - non-integral power of a matrix?
Keywords: matrix
Date: 13 Aug 92 04:18:24 GMT
Organization: None.  Santa Barbara, CA

In <Aug.10.15.45.34.1992.26563@clam.rutgers.edu>
gonzalez@clam.rutgers.edu (Ralph Gonzalez) writes:

>Hi.  Does anyone know of an algorithm to find a non-integral
>power of a matrix, e.g. A^.5 or A^1.3? Thus, A^2 is the same
>as AxA and A^0 is the identity.

At that level of generality, there is only one way:

1) diagonalize A:
   P^-1 * A * P = D,   where P nonsingular, D diagonal.
   The diagonal entries of D are the eigenvalues of A,
   the columns of P are the eigenvectors.  Get EISPACK
   or LAPACK.

2) massacre the diagonal entries of D:
   D'ii := Dii^r

3) un-diagonalize:
   A^r := P * D' * P^-1

This works for other functions besides just raising to a power.

>I imagine if such a thing is defined, then there are conditions
>on A...

You hope that the eigenvalues of A are real and positive.
Otherwise, you have to pick a sensible branch cut.

>The application is a matrix which represents a transformation
>which is applied to a 3D homogeneous coordinate vector, to 
>represent change per unit time. Then, if I want to find out
>where the coordinate moves to at time=t, I would multiply by
>the matrix A^t. Thus, at time=0 there is no change and at time=2
>the unit transformation matrix is applied twice...

Sounds like you want exp(A*t), but that's up to you.
-- 

--Pierre Asselin, Santa Barbara, California
  pa@verano.sba.ca.us
Newsgroups: sci.math
From: regeorge@magnus.acs.ohio-state.edu (Robert E George)
Subject: Re: Continued Fractions
Keywords: Help, Continued Fractions
Organization: The Ohio State University
Distribution: na
Date: Thu, 20 Aug 1992 19:12:23 GMT

There is a role played by continued fractions in approximation/numerical
methods, see for instance Numerical Analysis, by Ward Cheney and David
Kincaid (Brooks/Cole //1991), pp. 403ff. Other references are given
therein.

Robert George
(speaking only for myself)
From: davidc@panix.com (David B. Chorlian)
Newsgroups: sci.math
Subject: Re: Did Gauss ever say ...
Date: 21 Aug 92 01:39:46 GMT
Organization: PANIX Public Access Unix, NYC

"Mathematics is the queen of the sciences and arithmetic the queen
of mathematics... " is attributed to Gauss by
Sartorius von Waltershausen in "Gauss zum Gedaechtniss", Leipzig, 1856,
by Robert Edouard Moritz in "Memorabilia Mathematica", New York, 1914

David B. Chorlian
Neurodynamics Lab
SUNY/HSCB
Newsgroups: sci.math,sci.physics
From: israel@unixg.ubc.ca (Robert B. Israel)
Subject: Re: tensors
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Fri, 21 Aug 1992 01:15:08 GMT

In <schlegel.714343318@cwis> schlegel@cwis.unomaha.edu (Mark Schlegel) writes:

>frisinv@alleg.edu writes:

>>  I was reading a book on general relativity and the author began talking  
>>about tensor analysis, something I'd never heard of. I get the feeling  
>>tensors are some sort of number, but I'm not sure. Could anyone enlighten  
>>me as to what they are and how to use them?
>>----
>>Vince

>Tensors are very useful in physics and are generally the next level of
>complexity beyond vectors.  The simplest tensors are those of second rank
>which contain nine terms (usually scalars and vectors are not called tensors
>but they do belong to the tensor family) usually the name "tensor" refers
>to those of rank 2 and above but tensors generally as a class also
>contain scalars and vectors like this:

>a tensor has 3^N terms where N is the rank number

This is true if you are working in 3 dimensions (which the ordinary
physicist usually does, I guess).  In relativity we usually work in 4
dimensions, so it's 4^N.
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Newsgroups: sci.math
From: algebra@leland.Stanford.EDU (Leong Weng Ng)
Subject: Re: Still another problem.
Organization: DSG, Stanford University, CA 94305, USA
Date: Fri, 14 Aug 92 22:25:51 GMT

In article <1992Aug14.142149.16686@mcs.drexel.edu> dmagagno@mcs.drexel.edu (David Magagnosc) writes:
>
>This is a very nice proof, but I will say that I am a little surprised
>that no one has posted the (correct) value.  Since it follows very
>easily from this approach, permit me to summarize.

I was the only person who posted a value, which is incorrect.
Upon differentiating my expression again, I saw that it is
not so simple. Thanks for the hint. While I'm at it, I also
posted an incorrect solution to Bill Taylor's quadrilateral
problem. That was due to carelessness in applying the 
determinant formula. Thanks to the fellow netter who pointed
that out and proposed a corrected solution by private mail.
It has been fun working on these puzzles. I will definitely
try harder to post a correct solution in future.

Weng Leong
Stanford U.


Newsgroups: sci.math
From: polani@Informatik.Mathematik.Uni-Mainz.DE (Daniel Polani)
Subject: algo. determining closed form integral from c.f. function 
Followup-To: Goat Puzzle
Organization: Computer Science, Johannes Gutenberg-Universitaet
Date: Thu, 13 Aug 1992 19:15:35 GMT

In article <4699@balrog.ctron.com>, wilson@ctron.com (David Wilson) writes:

   It occurs to me that there is an algorithm for determining if a closed-form
   function has a closed-form integral.  Would a varition of this algorithm
   apply to the grazing goat equation?

An algorithm which was said to do this, I have been told, has been devised by
Risch (sorry, I cannot be more precise) based on an idea of Liouville from the 
last century. That's all I know. 

Does anybody know the general idea used by this (or a similar) algorithm ?

-- 
Daniel Polani - polani@informatik.mathematik.uni-mainz.de
From: kraai4712@iscsvax.uni.edu
Newsgroups: sci.math
Subject: Re: Binomial like recurrence
Date: 13 Aug 92 09:28:18 -0500
Organization: University of Northern Iowa

In article <1992Aug12.171516.24574@menudo.uh.edu>, rmverma@cs.uh.edu (Dr. R.M. Verma) writes:
> In article <1992Aug12.100933.13400@aifh.ed.ac.uk> tw@aisb.ed.ac.uk (Toby Walsh) writes:
>>
>>Has the related recurrence,
>>
>>    u(n,k) = 2.u(n-1,k-1) + u(n-1,k)
>>
>>been studied? Does it have a closed form?
>>What is its asymptotic behaviour?
>>
>>And more generally, how about,
>>
>>    v(n,k) = a.v(n-1,k-1) + b.v(n-1,k)
>>
>>for integer a and b?
>>
>>toby
> 
> Yes, they have been studied and there are answers to some of your
> questions in the literature. 
> 
> See the paper by Monier in Journal of ALgorithms 1980, and
> Ch. 8 of The Analysis of Algorithms by Purdom and Brown (more references
> can be found here).

Please don't forget the wide range of interesting materials put out by the
Fibonacci Society.

> 
> I would like to know your reason for studying them.
> 
> R.M. Verma

I'd love to know also, this is near my favorite playground.

jim kraai
Newsgroups: sci.math
Subject: Math Gods & lower Saints
From: madams@ecst.csuchico.edu (Michael E. Adams)
Date: Sat, 22 Aug 1992 08:00:51 GMT
Distribution: usa
Organization: California State University, Chico

Question: given Monthly_Payment, Length_of_loan (in months), and the
          Principal,  what is the interest rate (compounded monthly)? 


Dear Math God:
   Do you have time to explain the solution of a small finance problem?
I'm trying to write a Pascal program that has a loan computation section.
There are lots of code examples for Amortization of a loan, and the normal
"time value of money" functions, but I want to solve for the intrest rate also.

Here is an example to expalin WHY I want to do this:

Let's say a car dealer is trying to set the terms of a loan to sell a car.
Given the following Information:
---------------------------------------
(principal)                   10000.00   Price set by owner
(interest rate APR)              10.50   Bank + dealer rate
(number of months)                  36   Banks Max loan period
(months between payments)            1

(payment)                       325.02   Amortization results
(sum of interests)             1700.88
---------------------------------------

The buyer says he likes the car, but can only make $300/month payments. Thus,
the dealer has a problem: The bank won't let him extend the loan period,
and the owners won't let him drop the price.  The dealer CAN lower his
portion of the interest rate, but how much should he lower it?
                                  ^^^^^^^^^^^^^^^^^^^^^^^^^^^
Thank you for you help.
-Michael

         (___)      |  Michael E. Adams
         (o o)      |  
  /-------\ /       |  P.O. Box 5027
 / |     ||O        |  Chico,  California  95927-5025    U.S.A.
*  ||,---||         |
   ~~    ~~         |  Internet: madams@cscihp.ecst.csuchico.edu
-- 
Hi! I am a .signature virus. Copy me into your .signature to join in!
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: Prime conjecture
Date: 13 Aug 92 23:27:36 GMT
	<18990@nntp_server.ems.cdc.com> <1992Aug12.214311.8476@wdl.loral.com>
Organization: M.I.T. Artificial Intelligence Lab.

When I was in high school, I was playing around with twin primes and
noticed that for the first few examples, the following was the case:

Let the lower member of the r-th pair of twin primes by the n-th prime.
Then if n is prime, it is the r-th prime. For example,

1 2 3 4 5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  27 
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103
  1 2   3      4        5        6           7        8                  9

When I  was an undergraduate, I showed this to some friends of mine who
checked it further on a computer and found that it breaks down fairly
quickly beyond that point. However, it is still reasonable to ask whether
it is approximately true. For example, does it seem to be consistent with
anything that is known about the bounds for the number of twin primes less
than x?

Any comments are welcome.

Allan Adler
ara@altdorf.ai.mit.edu