MFGoode Memorial Lecture: "Nbr Theory Since 1964"

[ewmayer]

This one was just too pithy to ignore:

Originally Posted by mfgoode

This post is not on algebra or geometry but to update it I would like to recommend a book, I refer to often and keep at my bedside.

'Recreations in the the theory of Numbers- The queen of mathematics entertains' by Albert H. Beiler is excellent. It gives the basic theory on Numbers and also solved problems which are many (100's appx.)

Yeah you guessed it. This is the book I'm taking my problems from!

It is a Dover publication and printed in 1964. Don't get put off by the year !

Its as modern as it could be as number theory has not changed much since then except some new stuff by Pomerance et al.

Thanks for that amazingly concise synoptic view of progress in the field, Mally. It's always so nice to have an expert in the field fill us, the Great Unwashed, in on the major developments.

I perhaps am being much too bold for my own good, but I would humbly like to help the rest of readership, most of whom are not as intimately acquainted with the research in the field as you, understand a bit more of the details of your marvelously terse "Pomerance et al". Some of the researchers who did major parts of the work in question might themselves be surprised to find out that they are in fact a mere et or al in a great all-encompassing "Pomerance et al". To them, I say: sorry folks, but sometimes the truth is painful. So, without further ado, just a small portion, call it installment 1, of "Pomerance et al": Some of the mere-footnotes-in-number-theoretic-history since 1964 include the following:

1) Discovery of some trivial but quirky connections between elliptic curves and modular forms (Taniyama-Shimura-Weil conjecture), one of whose trivial corollaries (but nonetheless of some interest to the anal-retentive numerati among us) was the proof of Fermat's Last Theorem by a certain Mr. Et Al Wiles. (From the name I think he's Israeli, but I don't know if he goes by "Et" or "Al" amongst his friends.)

2) At least 5 footnote-worthy factoring algorithms: CFRAC, QS/MPQS, P+-1, Elliptic Curve, NFS (Special and General). The et als were quite busy in this area.

3) Halfway-decently-fast algorithms for large-integer arithmetic, especially the FFT-based Schönhage-Strassen algorithm on which nearly all modern huge-int arithmetic implementations are based. (IIRC, That's Et Schönhage and Al Strassen.)

4) Several Number theoretic cryptosystems, for instance generic publ,ic-key PGP, also RSA and ECC, none of which are as secure as the WW2 Enigma, but which happen to work better on e-mail.

5) Several almost-as-good-as-intelligent-guesswork general-purpose primality proving algorithms, starting with APRCL and ECPP (both not 100% deterministic) to the recent AKS, the first fully deterministic polynomial-time primality proving algorithm.

..and some minor theoretical developments which I'll let the et al theoreticians fill us in on.

==========

[Edit: Alright, who's the disrespectful back-bencher who sent me the anonymous "Guest" PM suggesting I add "So Mally, maybe it's just *your* number theory has not changed much since 1964... (and I'm being generous by at least 300 years, most likely.)" to this posting? C'mon, 'fess up!]

[mfgoode]

Dear Ernst, From the outset let me categorically state that I do not disagree with you.

However since you have gone ballistic over my innocent use of the expression ‘et al’ I think your detailed post on recent mathematical advances comes under what the Germans call ‘SCHADENFREUDE’ the English equivalent being ‘Roman Holiday’

Since Alan Matheson Turing in 1936 brought out his Turing test and Turing machine followed up by the logician Alonzo Church, mathematics took a sudden twist with the assault on the illusory goal of absolute mathematical Knowledge.
They concluded that there is simply no recipe for finding arithmetic truth (I would add by computer)

Since 1966 when the Chinese math’cian Chen Jing-run came up with his proof about sufficiently large even numbers and a weaker version of Goldbach’s conjecture it has been considered as the most significant contribution to prime number theory in the past 4 decades (Hoffman).

In brief, mathematical theory has not progressed as fast as the computational arithmetic which has moved in leaps and bounds.

Yes the computer has proved useful to math’cians finding large primes, solving Archimedes’ cattle problem, breaking codes, proving the 4 colour map theorem and discovering new shapes. Nevertheless there are limits to what the computer can do.

As the celebrated author Paulo Ribenboim in his very preface of his book ‘The Little Book of Bigger primes’ [TLBBP] states that “limited progress was made in the theoretical results. The old classical problems remain open and continue defying our great minds”

Since 1984 the last 5 popular publications by Schroeder M.R., Crandall R.E., Bach E & shallit, Nakiewiez W., Crandall R. and Pomerance C. deal with science and communication, Algorithmic Number theory, Prime number theory and computational perspective.

Leaving me aside, I wonder what the Math’s Greats Euler and Gauss to name a few, would say about this type of mathematics and I leave the question open.

To return to your detailed post I do not accept Andrew Wiles some 140 page proof as the ultimate. I also think that your remarks on him are derogatory to the British People.

Fermat had a short proof which is yet to be discovered by any today. I stick my neck out that a concise proof will be found, and that no amount of computation can prove it. Similar is the Riemann Hypothesis or for that matter the Beal’s conjecture!

As for myself like chess grand masters and ballet stars I fear the day when I ‘sense my abilities slipping away and that day usually comes along before old age’ (Hoffman)

As Paul Erdos puts it quoting his friend Ulam “The first sign of senility is that a man forgets his theorems, the second sign is that he forgets to zip up and the 3rd sign is he forgets to zip down”!

Incidentally since you are so well informed I think you should have mentioned our own colleague Paul Leyland (Xilman) in his factorization of Cullen and Woodall numbers.
His work is worthy to have been mentioned in Paulo’s book [TLBBP] on page 242.

In conclusion Prime Number theory is always there to stay but the primes have become bigger and bigger!

Mally

[hhh]

Well, the theory is the same as 1964, I guess. There has been development (some?), but what was true in 1964 still is, and will continue to be so, forever and ever and evar.

Ha! Until some mfgoode Memorial Number Theory Bronze Medalist proves you all wrong. You will see, DOOM is impending, the end of the World near! All the mathemathics has to be rewritten, in order to purge the inflation!!
The only way to purify yourselves is to drink a lot of . Them is them only escape from fate them. And
H.

[bdodson]

Originally Posted by mfgoode

Dear Ernst, From the outset let me categorically state that I do not disagree with you.

However since you have gone ballistic over my innocent use of the expression ‘et al’ ...

To return to your detailed post I do not accept Andrew Wiles some 140 page proof as the ultimate. ...

Fermat had a short proof ... Mally

"Et. Al." isn't the issue. Ribenboim seems to have a somewhat different
view; a scan through his recent publications brings up a book written
in 1999 (five years after Wiles' proof; some 350 after Fermat's). The
title is

Fermat's last theorem for amateurs. Springer-Verlag, New York, 1999.
xiv+407 pp. ISBN: 0-387-98508-5

The math review notes that the book ends with "a few words" about
Wiles proof, and includes the comment

This book would likely be of great interest to an enthusiastic undergraduate
with a basic knowledge of rings and fields. In addition to describing the
history of one of the great problems in number theory, the book provides a
gentle and well-motivated introduction to some important ideas in modern
number theory, ...

Unless Ribenboim's editors have distorted his intentions, even amateurs
are expected to at least have an interest in modern abstract algebra.
In any case, the above citation indicates that Prof Ribenboim has written
over 400-pages; which would seem to be an indication that he didn't
consider Fermat's short proof to have settled the issue. In fact,
Ribenboim seems a bit obsessive about this, scrolling down the reviews
a bit, there's a 1993 paper with a review that mentions

This paper is very much in the spirit of an after dinner talk. (In fact, it was. ...)

The title? "Fermat's last theorem, before June 23, 1993." Some more from
the review provides the info that the date was when Wiles first announced
that he had a proof; and that the lecture/notes were "before September 19,
1994, the date on which Wiles came "suddenly to a marvelous revelation"
and "saw in a flash ..." (how to complete the proof)". Two more, even
older, papers on "the first case" of Fermat's Theorem; then there's
a paper "Kummer's ideas on Fermat's Theorem" from the beginnings of
the subject of algebraic number theory. Several more on those two
topics. Ah, here's the one I was looking for, his famous one,
13 lectures on Fermat's last theorem. Springer-Verlag, New York-Heidelberg,
1979. xvi+302 pp. ISBN: 0-387-90432-8. Another 300 pages, yes?

So anyway, I consider myself fairly fortunate that have the opportunity
to hear several lectures on recent number theory this spring. There were
three here this spring from Prof Andrews on his discovery of the "lost
notebook" of Ramanujan, including references to recent results with his
co-authors Berndt and Ono that go on to complete (in theory, at least)
the congruences on partition numbers. I checked an undergrad summer
research paper written with Ono's direction; not only congrueces
mod 5, mod 7 and mod 11 of the lost notebook, but way new ones,
mod 13, 17, 19, 23 and 31. The math? Coefficients of power series
from gamma-0-572, using Hecke operators, which --- were studied
in the 30s and 50s (Hecke, Hasse, Taniyama, ...), but are most
recognized by non-amatuers as having occurred more recently in
Wiles' proof.

Two more, from the Clay Institute research awards. In a session
moderated by Andrew (no "s" this time), one describing consequences
of the solution of Serre's conjecture, including that higher dimensional
abelian varietes over Q (not just dim 1, elliptic curves; but with real
multiplications) are also modular (gamma-0, again). The second proving
Sato-Tate, describing the distribution of solutions of elliptic curves
mod p (as in ECC and ECM). Both using methods introduced in Mazur
and Wiles; the latter going well beyond GL2 and GL3 (Fermat), perhaps
suggesting that GLN is within reach of current methods.

Well, I've heard before from Mally on previous posts from me (not
impressed by my jargon; nor my near 30-years of introducing undergrads
to modern algebra and computation). I did my best to resist ewmayer's
redirection of the credit given to Carl for post-1964 number theory.
Looks like hhh's gotten in ahead of me. -bd

[mfgoode]

Originally Posted by hhh

Well, the theory is the same as 1964, I guess. There has been development (some?), but what was true in 1964 still is, and will continue to be so, forever and ever and evar.

Ha! Until some mfgoode Memorial Number Theory Bronze Medalist proves you all wrong. You will see, DOOM is impending, the end of the World near! All the mathemathics has to be rewritten, in order to purge the inflation!!
The only way to purify yourselves is to drink a lot of . Them is them only escape from fate them. And
H.

Ha! Ha! Brilliant hhh!. So you agree with me as to the theoretical aspects of mathematics. No its disappointing there are No Euler's or Gauss's nowadays. There is too much dependence on computers unfortunately!

I knew a guy, who when pocket calculators first came out, got so used to their use, that even if you asked him whats 2 x 2 he would reach for his cal. Its like a bullfighter in the ring reaching for his sword to plunge into the defenceless bull at every opportunity !

Well I am not out to prove any one wrong or any method wrong. I am here in my short life, of what is left, to tell the truth and make it plain and simple to others and to learn more and more of that truth from those who tell it. And what better than the medium of Maths?

I better end as this might be too tedious to you to read/translate.

Thanks all the same for your comments though.

Mally

P.S. I have cut my cigs from 40 to 20 a day but my consumption of coffee is the same.

[davieddy]

Originally Posted by mfgoode

P.S. I have cut my cigs from 40 to 20 a day but my consumption of coffee is the same.

I have cut my tobacco consumption from 1 oz per two days
to three days. I have also reduced my coffee consumption.

OTOH I drink more cider. Do you drink Mally?

David

[mfgoode]

Originally Posted by bdodson

"Et. Al." isn't the issue. Ribenboim seems to have a somewhat different
view; a scan through his recent publications brings up a book written
in 1999 (five years after Wiles' proof; some 350 after Fermat's). The
title is
~ ~ ~
Well, I've heard before from Mally on previous posts from me (not
impressed by my jargon; nor my near 30-years of introducing undergrads
to modern algebra and computation). I did my best to resist ewmayer's
redirection of the credit given to Carl for post-1964 number theory.
Looks like hhh's gotten in ahead of me. -bd

Thank you bd for your detailed post. I'm glad that it spurred you onto taking a definite stand on Paulo by the many reviews you have cited.
Well in my post I did not only refer to Ribenboim but also Paul Hoffman A H Beiler etc.
To be honest at the time of writing my post I did not remember any of the modern contributors except Carl Pomerance as I was forwarded a personal communication of his hence the et.al. for the others.

Originally Posted by bdodson

Well, I've heard before from Mally on previous posts from me (not
impressed by my jargon; nor my near 30-years of introducing undergrads
to modern algebra and computation).

Well bd I honestly dont remember answering or noting any of your replies. As you can see that the posts I tackle are few and not on programming etc.

If you dont reveal yourself in your posts how am I to know of your work and experience? I have appealed to the powers that be to make it mandatory to mention qualifications, experience and last or present job held and country of living, in their Profiles. Because of an international character of the forum it will also enable us to subdue any racism that may be inherent in us with a better understanding of countries and climes.
That will give one an idea as to whom you are addressing a post too.

Well with this current post I feel I've got to know you better bd.

30 years ? Welcome to the senior wranglers club!

Abstract Algebra? I have a book I refer to at times mainly as a reference.
It is 'Contemporary Abstract Algebra' by Joseph A. Gallian. Apart from the maths it has short biographical sketches of the great masters. This seems to be the trend of publications these days. Please combine your lectures with a historical background of the masters. I am a firm believer in studying the masters first and foremost.

You are probably on the faculty of Lehigh

I envy you attending all those seminars by world famous pioneers in their subjects.

I wish you enlightenment in your forthcoming seminars.

Regards,
Mally

[mfgoode]

Originally Posted by davieddy

I have cut my tobacco consumption from 1 oz per two days
to three days. I have also reduced my coffee consumption.

OTOH I drink more cider. Do you drink Mally?

David

Your consumption of tobacco is quite high. Do you smoke a pipe or roll your cigarettes? If the former then I request Xyzzy to change your avatar to a white coated professor smoking a Sherlock Holmes curved pipe to make it look like a sax as you are a lover of jazz thus combining both traits.

To the second question. Yes I do, but socially esp. when my sons visit me to keep up with the modern generation. I have a stock though and due to economic reasons I have switched to Red from Black Label. I am retired 10 years now!

Mind you most docs recommend two alcoholic drinks a day to keep the heart and memory going. Coffee is also good for the heart and smoking for the nerves.

So to those who flame us lets light up the peace pipe! Try a 'hookar' once in a while It's very cooling.

Mally

[xilman]

Originally Posted by mfgoode

:
If you dont reveal yourself in your posts how am I to know of your work and experience?

I would have thought that Bruce "reveals himself" adequately well in his posts here.

How to know: have you considered asking Google? Or even any of several here who apparently know him quite well, based on our responses to his posts?


Paul

[mfgoode]

Originally Posted by xilman

I would have thought that Bruce "reveals himself" adequately well in his posts here.

How to know: have you considered asking Google? Or even any of several here who apparently know him quite well, based on our responses to his posts?


Paul

Well Paul I have made enough of a veiled apology to Mr. bdobson in my post.
This is the first time he has replied directly to me and so I answered truthfully that I am not aware of his posts as I dont read thru each and every post on the forum.

Originally Posted by Originally Posted by bdodson

Well, I've heard before from Mally on previous posts from me (not
impressed by my jargon; nor my near 30-years of introducing undergrads
to modern algebra and computation).

Self praise needs no recommendation!

He has written three threads and the one in the lounge I read thru when there was a dialogue between him and davieddy. I didnt think it was worth commenting on, even though he mentions an obscure Indian mathematician. That was the first time when his name registered in my brain.

Originally Posted by xilman

How to know: have you considered asking Google? Or even any of several here who apparently know him quite well, based on our responses to his posts?

I dont think I have to go to that extent to ask 'the several here' to form an opinion on him. I have met David Attenborough have you ? And is it necessary for you to know someone I know?

Well not to be stubborn I googled for bdodson as you advised. The query was 'do you mean dodson?' So I confirmed that and got a whole list of firms dealing in aircraft and misc. parts. There was a professor called C dodson which does not match.
Since he signed of bd/ I was not familiar with his first name Bruce. My elder son is named Bruce and I would have remembered it if he had,. having been 35 years in the hospitality business and HRD

To play the game further I googled his full name and so got his particulars.

Then I googled xilman to know more about you. All I can say is that you are a very mysterious character, a former member of Slashdot

Well I have no grudge about Bruce's post. I am delighted that he saw fit to reply to my post.

And in closing let me tell you Paul that I am no hero or buddy worshipper!
I worship the Creator and not the creature in His creation

Mally

[xilman]

Originally Posted by mfgoode

I dont think I have to go to that extent to ask 'the several here' to form an opinion on him. I have met David Attenborough have you ?

Nope. If you had given an indication beforehand that you had done so, and if I had wanted to find out more about him as a person, I may very well have asked you.
If we're in the name-dropping business, have you met Stephen Hawking? I've had dinner with him on two occasions and, for that matter, man-handled his wheel chair up some inconvenient steps.

Originally Posted by mfgoode

Well not to be stubborn I googled for bdodson as you advised. The query was 'do you mean dodson?'

The seach ""bdodson factoring" turms up a pageful of relevant links, including a few posts on the Mersenne forum and a link directly to Bruce's home page. Anyway, just because Google thinks you can't spell doesn't mean that it invariably knows better than you.


Paul