HDTenEightyP/NetNews huggingface dataset:
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: HELP on RSA scheme for Encryption/Decryption
Organization: U of Cambridge, England
Distribution: na
Date: Wed, 11 Nov 1992 17:01:56 GMT

In article <6605@dciem.dciem.dnd.ca> jorchard@dretor.dciem.dnd.ca writes:

> I would like to know the algorithm for encrypting/decrypting codes
> with the RSA scheme.  I took it in first year algebra, but I sold my
> textbook and can't remember for the life of me how it went.
>
> Could someone out there with better resources (or better memory)
> help me?

Surely.

Everyone who wants to be sent messages using the RSA scheme picks two
large prime numbers, say p and q, and multiplies them together to give
a number n=pq.
He also picks a number d coprime to phi(n)=(p-1)(q-1).
He computes another number e, which has the property: de congruent 1 mod phi(n)
He makes public the numbers n and e.
He keeps private the numbers p and q; also phi(n) [as given pq and (p-1)(q-1)
one can find p and q by solving a quadratic equation]; also d [for reasons
which will become clear].

Now, the first important fact you need to know is that any number coprime to n
has a^phi(n) congruent to 1 mod n.

So, if someone wants to send this guy a message, she does this: encode the
message in some standard way as a sequence of numbers less than n and coprime
to n. Then for each such number a, compute a^e mod n; the resulting sequence
is the encrypted message.

To decrypt this, suppose b is one of the numbers produced; then compute b^d
mod n; this is a^(de), which is a^(something times phi(n) plus 1), which is
a^(something times phi(n)) times a, which is a. [All this is done modulo n.]

This is why we can't make d public, of course: it would enable everyone to
decrypt messages.

Now, how secure is this? Well, basically the only way to crack it is to factor
n; that is, to find p and q. It is thought (though not proved) that it is hard
to factorise numbers; that is, that provided some precautions are taken in
choosing p and q there is no quick way to find what the factors of pq are.

Precautions include: not having p,q too close together; making sure p-1 and q-1
don't have any really small factors; probably lots of other things I can't
think of offhand.

I hope this is of some use. If I've assumed you know more mathematics than you
actually do, send me some e-mail or something and I can explain what the words
mean...

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
From: hdev@dutiaj.tudelft.nl (Hans de Vreught)
Newsgroups: sci.math
Subject: Re: Relatives with Crackpot Proofs
Date: 12 Nov 92 09:05:33 GMT

RVESTERM@vma.cc.nd.edu writes:

>why don't you just tell him where his proof went wrong? sheesh.

>just say, "this step here is wrong, because blah blah blah."
>you don't have to be derisive about it, and tell the guy how
>dumb his proof is.  just tell him why it's wrong.

Ah, but that's often the problem. If you have a paper of a well trained
mathematician, you will find it is nicely structured. You can read the proof,
and that you can sometimes spot the error. If you can see the error and you
can explain it to him, fine...

But we were talking about Crackpot Proofs. Something like a paper of 20 pages,
with no beginnning or end. If you read it you have no idea where the error is.
Even here at the univ. I sometimes see examination papers from students where
they give "proof" of something which is almost impossible to follow. But since
it is an examination paper you must review it. But still about 1 or 2 papers
are so misty, that they get some credit because you were unable to spot the
error. And students have some training in math.

And that was probably the problem of the original author: the paper didn't
make sense or he couldn't understand the paper (probably already lost on page
1). Spotting errors in such things isn't always easy (even professional
mathematicians make errors that are hard to find). And then you have to
convince such a person what is wrong, and then he starts to say "No, you are
wrong, because <clearly false> is true.". Some people can be really pigheaded,
defending every little detail as if their life depend on it.

Just telling a relative his proof is dumb? You've gotta be kidding. It would
give very cross eyes and you will get some very nasty remarks from your
relatives asking you why you think you're so much better than him (right that
was not what you said, but that still probably what they make of it). You
must still live with your relative and your other relatives. So a little
diplomacy is on its place.
--
Hans de Vreught                          | John von Neumann:
hdev@dutiaa.tudelft.nl                   | Young man, in mathematics
Delft University of Technology (TWI-ThI) | you don't understand things,
The Netherlands                          | you just get used to them.
From: dvf@kepler.unh.edu (David V Feldman)
Newsgroups: sci.math
Subject: Re: Univariate polynomial equations and the FAQ
Date: 9 Nov 1992 05:12:34 GMT
Organization: University of New Hampshire  -  Durham, NH
NNTP-Posting-Host: kepler.unh.edu

In article <1992Nov6.184527.20793@sun0.urz.uni-heidelberg.de> gsmith@clio.uucp (Eugen W. Schmidt) writes:
>In article <1d72mnINNq2p@mozz.unh.edu> dvf@kepler.unh.edu (David V Feldman) writes:
>
>>Fix an integer m.  Let K be the extension of Q obtained by adjoining
>>all roots of all polynomials of the form
>>     n     m    
>>	  x + a x + ... a
>>		   m         0
>
>>where the coefficients are rational.
>
>This sounds like the algebraic closure of Q, Q-bar.

While it is true that I did not ask the question that I intended to,
I don't see any reason that  K = Q-bar.  Remember that  m  is fixed.
So I was adjoining roots of polynomials that, for large  n , would
have very few non-zero coefficients.  Actually the field  K  that
I intend is even larger.  Specifically, what I should have said is:

Fix an integer  m .  Let K be the smallest extension of Q closed under
the adjunction of all roots of all polynomials of the form

	   n     m
      x + a x + ... a
		   m         0

with the a_i in K .  Then let s(m) be the smallest degree of a polynomial
with coefficients in  Q  and no root in K.  Then if  m=0, s(0)=5 by
Galois theory.  Again, the question is, what is known about s(m), m>0?

>   Eugen W. Schmidt/Der Brahms Gang/IWR/Ruprecht-Karls University 
>               gsmith@kalliope.iwr.uni-heidelberg.de

David Feldman
Xref: sparky sci.math:14872 misc.education:4270
From: uejiowh@dssv01.crd.ge.com (W. H. Uejio)
Newsgroups: sci.math,misc.education
Subject: Re: Is Math Hard?
Date: 12 Nov 92 18:39:34 GMT
Reply-To: uejiowh@crd.ge.com
Organization: GE R&D, DICE Program

My daughter's fourth grade class (she is a student) has a very
good integrated course plan.  The teacher, IMHO, has done a great
job of integrating multiple subjects into a single theme which 
makes the students understand, learn, and problem solve.

The major topic is the State of New York.  He built a huge topographical
map of New York. They started by seeing how the ice age (history and 
science) created certain lakes, rivers, mountains, etc. They calculated
how much material was moved over time (math).  

He also broke up the class into the regional voting sections, similar
to the state legislature and have them proposing and voting on things.
They now see how the City of New York affects them, even though
they are 150 miles away. (New York City voted to have a hazardous waste
dump in upper New York).

Later in the year, he will have the map contain the forest (he will include
the Indians and Colonial period etc.) and the 
loss of forest with the re-growth in the 1930's.  He also will have the
children, after an electricity course in science wire the map showing how
the water and power from the entire state is sent to the City.

As part of this, there will be the calculation of the amount of water,
power, and losses involved in supporting New York city.

My daughter is finding it interesting and is far more interested
in current events because of this.  She was always a good math student
but now is starting to understand some applied math also.

Wayne Uejio             General Electric CRD   Disclaimer: opinions 
uejiowh@crd.ge.com      KWC-275                expressed are my own 
Office: 518-387-6441    P.O. Box 8             and not the company's
Fax:    518-387-7080    Schenectady NY 12301   opinion
Xref: sparky sci.math:14608 misc.education:4196
From: landsbur@troi.cc.rochester.edu (S. Landsburg)
Newsgroups: sci.math,misc.education
Subject: Re: Is Math Hard?
Date: 8 Nov 92 20:56:52 GMT
Organization: University of Rochester (Rochester, NY)

In article <1992Nov8.144030.3507@ms.uky.edu> cyeomans@ms.uky.edu (Charles Yeomans) writes:


>There are universities in the US which will take students without a 
>high-school diploma - but probably only the good ones.  I would 
>imagine that it would be easier to get into Harvard than into **SU
>without a diploma.
>

I have one sad piece of confirmatory evidence.  When I taught first-year
graduate algebra at Colorado State two years ago, I had a Chinese student
who was one of two clear-cut "A" students out of 14 in the class.  In 
the middle of the semester, a dean with nothing better to do discovered
that her bachelor's degree from China was not technically complete, for
reasons having to do with the events in Tienanmen Square.  His initial
response was to insist that she be sent back to China immediately.  After
much cajoling, he allowed her to finish the semester first, but insisted
that she return after that.  I can not imagine anything like this 
happening at Harvard.
.
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: group theory for HS students
Date: 8 Nov 92 16:01:34 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: ara@zurich.ai.mit.edu (Allan Adler)

In article <ARA.92Nov6191321@camelot.ai.mit.edu>, ara@zurich (Allan Adler) writes:
>I would be interested in having the names and authors and publishers
>of books that teach abstract algebra and which say explicitly that they
>are aimed at high school students.

There are not too many that can even come close.  G Papy GROUPS is an
obscure book, with lots of color to illustrate symmetry.  Presumably
some of the Rubik's cube books may have what you want.  A motivated
high school student should be able to follow Fraleigh's text.

There is also, off to the side, Knuth SURREAL NUMBERS and E Landau
FOUNDATIONS OF ANALYSIS, and Lang's two Springer-Verlag paperbacks.

And read the introduction to Dieudonne LINEAR ALGEBRA AND GEOMETRY.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
From: jack@dcs.glasgow.ac.uk (Jack Campin)
Newsgroups: sci.math
Subject: Re: Michael Spivak
Date: 6 Nov 92 14:57:00 GMT
Reply-To: jack@dcs.glasgow.ac.uk (Jack Campin)
Organization: COMANDOS Project, Glesga Yoonie

palais@binah.cc.brandeis.edu wrote:
> A number of people have asked how to order Spivak's "Calculus".
> It is published by his own company, Publish or Perish Inc.

> Publish or Perish Inc.
> 3701 West Alabama 450-130
> Houston, TX,   77027
> 713  524  5515 

What else do they publish?  All I've seen is Spivak's own huge 5-volume
differential geometry notes (this must be the biggest single-author
mathematics text ever written).

I've got the 1967 W.A. Benjamin edition of "Calculus", and the answers
volume - is the 1980 Publish or Perish one any different?  Why did W.A.
Benjamin drop it?

-- 
--  Jack Campin    room G092, Computing Science Department, Glasgow University,
17 Lilybank Gardens, Glasgow G12 8RZ, Scotland   TEL: 041 339 8855 x6854 (work)
INTERNET: jack@dcs.glasgow.ac.uk or via nsfnet-relay.ac.uk    FAX: 041 330 4913
BANG!net: via mcsun and uknet     BITNET: via UKACRL    UUCP: jack@glasgow.uucp
Newsgroups: sci.math
From: mitch@clsi.COM (Mitchell N. Perilstein)
Subject: Re: Three-sided coin
Organization: CAD Language Systems, Inc.
Date: 10 Nov 92 11:00:45

There are other three-sided objects than coins (right-angle cylinders).

Consider the following object. It's a cylindrical section, but best
described by its three orthographic projections: triangle, circle, and
parabola.  Another description would be a right-angle cylinder viewed
from the side (rectangle view) and bisected by the plane perpendicular
to the viewer and passing through one apparent corner of the cylinder to
its opposite; one half of the result is the object.

This flipping object can rest on one of two flat faces, or on its curved
side. So does anyone want to figure the angle versus radius to get a 1/3
randomizer?

---
 Mitchell N. Perilstein                        
 CAD Language Systems, Inc.                 Innovate, don't litigate.
 410-992-5700                               League for Programming Freedom
Newsgroups: sci.math
From: hamid@imada.ou.dk (Hamid Sarabi)
Subject: Bounding cylinder
Summary: find bounding cylinder for n points
Keywords: bounding cylinder 
Organization: Dept. of Math. & Computer Science, Odense University, Denmark
Date: Wed, 11 Nov 1992 20:36:39 GMT

Can anyone lead me to a library (PD or shareware) that will allow me to
compute the bounding cylinder for n points (polyhedra) in R^3.
In particular each polyhera should be bounded by either spheres or cylinders.

	Thanking you in advance.
================================================================
Hamid Sarabi
Denmark-Odense University
Email: hamid@imada.ou.dk
===============================================================
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Assorted questions and problems
Date: 10 Nov 92 14:21:16 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)

In article <1992Nov9.232300.18490@CSD-NewsHost.Stanford.EDU>, pratt@Sunburn (Vaughan R. Pratt) writes:
>The back-and-forth argument shows that there are exactly four
>*countable* dense chains, depending on whether they have a top and
>independently a bottom.  [...]

Yeah, yeah, enough corrections everybody.  I now think I was thinking
of the universal Boolean algebra result--I know for certain I once
looked at somebody else's paper of Shelah's proof that CH is needed.

Anyway, to whomever first raised the topic--look for Rosenstein
LINEAR ORDERINGS.  It's in the Academic Press green series.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Newsgroups: sci.math
From: jwr@lentil.princeton.edu (Jaroslaw Tomasz Wroblewski)
Subject: Re: Extended Fermat primes
Organization: Princeton University
Date: Thu, 12 Nov 1992 15:56:54 GMT

In article <BxKLzE.Bu9@cs.columbia.edu> kasdan@cs.columbia.edu (John Kasdan) writes:
>In article <1992Nov8.004737.13519@Princeton.EDU> tao@fine.princeton.edu (Terry Tao) writes:
>>
>>....   And it is highly likely that there are infinitely many primes
>>of the form n^4 + 1, n^8 + 1, etc. on the grounds that any polynomial which
>>is not factorizable should give infinitely many primes.  (Is there a name
>>for this conjecture? if you know it could you email me?)
>>
>
>I doubt that there is a name for the conjecture in that form, because
>it is obviously false.  Consider x^2 + x + 2.
>

There is a conjecture known as Sierpinski-Schinzel conjecture H stating the
following:

Let irreducible polynomials f1,f2,...,fn have the property
that there is no m > 1 such that for every x 
m | f1(x)f2(x)...fn(x)  

Then there exist infinitely many numbers x such that all 
f1(x) , f2(x) , ... , fn(x) are prime.

It follows from the above conjecture that there are infinitely many twin or
quadruple primes.


The conjecture has been formulated in 

Sur certaines hypotheses concernant les nombres premiers, Acta Arith. 4 (1958)
pp 185-208.

Note that there is a misprint there as the essential word "irreducible" has
been skipped. The correction appears ibidem 5 (1959) p 259.
--

Jarek (Jaroslaw Tomasz Wroblewski) ,   E-mail jwr@math.Princeton.EDU
Newsgroups: sci.math
From: tycchow@riesz.mit.edu (Timothy Y. Chow)
Subject: Re: Trivial! (was: Re: Help X^2 == Y mod N)
Organization: None.  This saves me from writing a disclaimer.
Date: Thu, 5 Nov 92 17:09:45 GMT

In article <1992Nov5.031326.11279@CSD-NewsHost.Stanford.EDU>
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>Wiener wandered away in deep thought and returned 20 minutes
>later to announce triumphantly that it was indeed trivial.

I've heard this exact same story told about Hardy and von Neumann and
various other mathematicians.  Is there any hope of figuring out if such
an event actually occurred and if so who it was?
-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949
Newsgroups: sci.math
From: darknite@camelot.bradley.edu (John S. Novak III)
Newsgroups: sci.math
Subject: Re: who should teach calculus?
Date: 12 Nov 92 05:42:22 GMT
Article-I.D.: camelot.darknite.721546942
Organization: Bradley University

Todd E. Peterson writes:

>I am interested in hearing net.opinions on the following:

>Is it a good or bad practice to use non-mathematicians to
>teach mathematics courses?  In particular, what about the
>use of engineering faculty to teach engineering calculus
>courses, and other applied mathematics courses?

I was taught differential equations, various bits of transform
theory and applications (LaPlace, Fourier, Z, discrete Fourier),
and probability theory by engineers, and had no problems with
them.

Then again, I shudder at calling an engineer a
'non-mathematician.'  As an electrical engineering undergrad, a
very _large_ part of my working knowledge base is mathematics.

--
John S. Novak, III		darknite@camelot.bradley.edu
Newsgroups: sci.math
From: tao@fine.princeton.edu (Terry Tao)
Subject: Re: Extended Fermat primes
Organization: Princeton University
Date: Sun, 8 Nov 1992 00:47:37 GMT

In article <1992Nov7.172207.17207@husc15.harvard.edu> blom@husc15.harvard.edu writes:
>Fermat primes have the form
>
>  n
> 2
>2  + 1.
>
>One extension of this form could be
>
>     n
>    2
>(2a)  + 1.
>
>The largest prime of this form I have encountered is
>
>    7
>   2
>120  + 1.
>
>Does anyone know of any larger primes of this form?

Aren't you just looking for primes of the form n^2 + 1?

There is an unsolved conjecture that there are infinitely many primes of
this form.  And it is highly likely that there are infinitely many primes
of the form n^4 + 1, n^8 + 1, etc. on the grounds that any polynomial which
is not factorizable should give infinitely many primes.  (Is there a name
for this conjecture? if you know it could you email me?)

	Terry 
Xref: sparky sci.math:14719 misc.education:4230
From: cornwall@bach.udel.edu (Ray J Cornwall)
Subject: Re: a first year grad student freaks out.
Organization: University of Delaware
Date: Thu, 12 Nov 1992 00:20:14 GMT

John C. Baez writes:
>That's why I'm glad that starting in high school I started trying to
>come up with math topics to do research on.  One has to face the unknown
>eventually if one is going to do research, so one might as well get used
>to it as early as possible (even if ones first research projects are a
>little silly).

For those of us who wish to do research (cause we'll get our names in
really cool journals! :) can you give us some pointers on how to come
up with such topics?
I've been trying to be more aggressive with my studies in math, to
look for such ideas, but I have a feeling I'm barking up the wrong
tree looking in textbooks and looking for my favorite topics here.
Where are some good sources of information, and what are some good
questions to ask myself?
(I know, Bad English on the last part...sorry...)
Ray Cornwall
University of Delawhere

Newsgroups: sci.math
From: pgolden@nhgs.vak12ed.edu (Patrick S. Golden)
Subject: Free Middle/High School Broadcasts
Organization: Virginia's Public Education Network (Hampton)
Date: Fri, 13 Nov 92 15:57:31 GMT

REACHING FOR THE STARS

A free, nationwide distance-learning program designed to  
encourage young minority and female middle and high-school  
students to pursue careers in science and engineering

2:00 p.m. - 2:30 p.m. (ET)

December 16, 1992
February 10, 1993
March 17, 1993
April 7, 1993
May 12, 1993

Program Description:            	
Each of the 30-minute programs in the five-part interactive series  
will profile a different recipient of a Virginia Space Grant 
scholarship or  fellowship.  These  young minority and female 
researchers will describe what  motivated them to  pursue careers 
in the sciences and engineering, what their lives are  like at the  
college or university where they study, and what their career  
aspirations are.   Each researcher will also teach a basic science 
lesson and briefly  discuss the  relationship of that science lesson to 
their university research  project.  Viewers  will have the 
opportunity to ask questions of the young researchers  via 
telephone  during an interactive segment of the program.  Pre- and 
post- broadcast material  will be mailed to those registering for the 
series. 

Target Audience:  Middle school and high school students and  
other interested viewers

Program Objectives:

The series is designed to provide viewers with:

- An awareness of the career possibilities and rewards that exist for  
students --   particularly minorities and women -- in engineering 
and the  sciences.

- A knowledge of the scholastic paths students must travel to  
become engineers  and scientists.

- An understanding of the nature of five young minority and  
female researchers'  college experiences and what viewers will 
encounter if they attend  college and  major in the sciences.

- Knowledge of five basic science lessons and how those lessons are  
related to  more sophisticated research.

Reaching for the Stars is a cooperative educational project of the  
Virginia Space  Grant Consortium, Old Dominion University and 
the National  Aeronautics  and Space Administration.

For Registration Information:	
Old Dominion University's Academic Television Services at  (800)  
548-4807

VIRGINIA SPACE GRANT CONSORTIUM
2713-D Magruder Blvd.
Hampton, VA  23666
(804) 865-0726
E-mail: pgolden@nhgs.vak12ed.edu
-- 
|-------------------------------|
|++Patrick Golden++             | 
|Virginia Space Grant Consortium|
|pgolden@nhgs.vak12ed.edu       | 
Newsgroups: sci.math,sci.math.symbolic
From: victor@tuamotu.wri.com (Victor Adamchik)
Subject: Re: Help wanted in integration.
Organization: Wolfram Research, Inc.
Date: Wed, 11 Nov 1992 21:14:17 GMT

In article <1992Nov10.235258.8777@ac.dal.ca> cordes@ac.dal.ca (John Cordes)  
writes:
> To those who have followed this thread:
> 
>  Sorry! I believe there is one more correction to make, for the case d=c 
> only. See below.

<stuff deleted>

>  Alas, my algebra did go wrong after all. The corrected result (for the 
> case d = c, a > b) is as above but without the factor 31, thus:
> 
> (1/32) * Pi^2 * b / c^2  +  Pi^4 * b^2 * (a-b/3) / 8.
> 
>  This evaluates to 328.981001... for a = 4, b = 3, c = 2, and so is now in 
> agreement with the numerical integration.
> 
>   
>   John Cordes, Dept. of Physics,
>   Dalhousie University, Halifax, N.S.,
>   Canada  B3H 3J5       Internet: cordes@ac.dal.ca


I got the same result for d = c, a > b.
For b > a it is 

(a*Pi^2*(3 - 4*a^2*c^2*Pi^2 + 12*a*b*c^2*Pi^2))/(96*c^2)

So finally we have:

for integers a,b,c and d:

1. c != d

if 0 < a <= b then (a*(c^2 + d^2)*Pi^2)/(4*(c^2 - d^2)^2) 
if 0 < b <= a then (b*(c^2 + d^2)*Pi^2)/(4*(c^2 - d^2)^2) 

2. c == d

if 0 < a <= b then (a*Pi^2*(3 - 4*a^2*c^2*Pi^2 + 12*a*b*c^2*Pi^2))/(96*c^2)
if 0 < b <= a then (b*Pi^2*(3 - 4*b^2*c^2*Pi^2 + 12*a*b*c^2*Pi^2))/(96*c^2)

--
Victor Adamchik
victor@wri.com