| MathPages__Number_Theory Kevin Brown's collection of sci.math postings related to number theory topics. |
| MathWorld_Number_Theory Index to articles in Eric Weisstein's MathWorld in the area of number theory. |
| A_Mechanical_Proof_of_Quadratic_Reciprocity A paper by David M. Russinoff describing the use of the Boyer-Moore theorem prover in mechanically generating a proof of the Law of Quadratic Reciprocity. PS/PDF. |
| Noncommutative_Geometry,_Trace_Formulae_and_the_Zeroes_of_the_Riemann_Zeta_Function Lecture notes by Alain Connes. |
| Number_Theory Proofs along with equation solvers and graphical views written in Java. |
| Number_Theory Dave Rusin's guide to number theory. |
| Number_Theory_Foundation Aims to collect donations from supporters of number theory and to disburse these donations to encourage research. |
| Number_Theory_Glossary Compiled by Robert Campbell. |
| Online_Seminar__Galois_theory_of_p-Extensions Email seminar on "Galoissche Theorie der p-Erweiterungen" by Helmut Koch. |
| Other_Maths A wide selection of number theory topics compiled by Jon Perry. |
| Some_Highlights_of_Arithmetic_Combinatorics Lecture notes and resources on combinatorial number theory by Terence Tao. |
| Some_Number-Theoretical_Constants Products of rational functions of p over primes, computed by Gerhard Niklasch and Pieter Moree. |
| Somos_Polynomials Related to Somos sequences and elliptic theta functions. |
| The_Somos_Sequence_Site Web resources for information on Somos sequences and related topics such as elliptic divisibility sequences. |
| Square-free_Gaps Algorithm and source code for the calculation of square-free numbers and gaps. |
| Transcendental_Numbers Maple worksheets, lecture notes and links to other resources by John Cosgrove. |
| The_Valuation_Theory_Home_Page A forum for all mathematicians who work in valuation theory or apply valuation theoretical results in their own field of research. |
| Vignettes_on_Automorphic_and_Modular_forms,_Representations,_L-functions,_and_Number_Theory By Paul Garrett. |
| Visible_Structures_in_Number_Theory By Peter Borwein and Loki Jörgenson. Recognising number patterns visually. |
| The_Work_of_Robert_Langlands Thesis, papers, manuscripts, letters and bibliography. |
| World_Records_for_Numerical_Palindromes The 196 Palindrome Quest and The Most Delayed Palindromic Number, by Jason Doucette. |
| American_Association_of_Tissue_Banks Professional organization for accreditation and information. |
| Asymptote_Cool_Guide_to_Cryopreservation Manual of the general principles of cryobiology, especially regarding the preservation of sperm and embryos. Detailed but accessible to the educated layman. |
| BioLife_Solutions,_Inc_ Conducts research and provides hypothermic and cryopreservation solutions for cell and tissue preservation, including the HypoThermosol and CryoStor products. |
| Biothermal_Sciences_Laboratory,_U__of_Illinois_at_Chicago Research on cryobiology, heat shock, and fever. |
| Conservation_of_endangered_animals_by_using_sperm_and_embryo_freezing A project to preserve several endangered mammals of far northern Europe, using cryopreservation. At the University of Kuopio, Finland. |
| Cryobiology International journal of low temperature biology and medicine. Published by the Society for Cryobiology. |
| Cryobiology_and_Anhydrobiology_-_School_of_Physics,University_of_New_South_Wales,_Sydney_Australia Research and introduction by Associate professor Joe Wolfe |
| Cryobiology__Low_temperature_studies_of_biological_systems A readable summary of the field of cryobiology by a college biology student, but with some confused opinions about the intent of the field. |
| Cryolab_-_Roskilde_University Cryobiology laboratory at Roskilde University, Denmark. Concentration on cold adaptation in animals, with ongoing research in antifreeze proteins, microtubules, cryopreservation. |
| CryoLife,_Inc_ Manufacturer of medical tissues, heart valves, and oncology drugs. Shown are tissue services, research and company profile. |
| Cryopreservation_Application_Guide A detailed introduction to cell preservation technology from Nalgene Nunc. PDF file (249K) |
| Cryopreservation_Research Research on trehalose and cryopreservation at University of Wisconsin, Molecular Thermodynamics and Statistical Mechanics Group. Click on "Research." Includes a link to their nanolithography program |
| Cryosite A company which provides secure off-site, ultra-low temperature and cryogenic storage for biological specimens. In Sydney, Australia. |
| Frozen_Fruit_Fly_Embryos_Hatched An article from "Technical Highlights" in the Oak Ridge National Laboratory Review (Vol. 26, No. 1). Reports the research of Peter Mazur and others from 1992. |
| Genternity_LLC Proposed services to include DNA preserved before cremation funeral for genetic preservation and immortality. |
| International_Bioresearch_Solutions IBS is a non-profit national organization that works with organ procurement organizations to place organs that are not suitable for transplant with medical researchers. |
| INVAM_-_Freezing_germplasm_in_liquid_nitrogen Research suggestions from the INVAM Culture Collection at West Virginia University. |
| Ken_Storey\'s_Lab The "frozen frog" expert at Carleton University, Ottawa, Ontario, Canada. |
| Lyophilization_Services_of_New_England,_Inc_ Contract manufacturer providing lyophilization (freeze-drying) of products, both vial and bulk. In Manchester, New Hampshire. |
(before it was known) would they dismiss it for being
"too simple'' a claim and reply: "Physics is at a stage where all
simple truths have already been discovered. Worse, your hypothesis is not even a
whole line and could have been uttered by any 3rd grader. If you want to do real
Physics why not lookup the present journals on the topics that interest you and
see if you can contribute something to truly advanced discussions on the
issues'' -- almost could give someone the impression that the aim is to study
and build on what the people in the field have written rather than to seek
natural truths. Mathematicians are very interested in each others'
"mainstream'' theorems and conjectures (as they should be), but here they
are presented here with foundational facts that they have missed and
rather than be indifferent should they not clear up the issue with an open and
explicit response?
Theorem 
:
Let 
be a function which is either always positive or always negative for all
integers 
,
where 
is a positive integer. Further let
,
where 
and 
are polynomials of 
with rational coefficients and let
be convergent series. Then the series converges to a rational number if and only
if f(n) can be broken into partial fractions
such that
for some integers
.
(The actual Theorem 
will give an infinite series converging to an irrational number
whereas
will give one going to an rational number. Also
is an integer and 
will give an infinite series that converges to an irrational number.
Around 1990 I had circulated the above form of the Theorem 2 by email
(with note that the actual theorem is slightly different) to some mathematicians
and most asked me to send them the paper. Mostly there was no further response
and if I pursued (such as with a phone call) I was often told some version of
"It is very difficult to comment on something you do not understand'' (this
quote is from an email).
(I had earlier mentioned the names of “two mathematicians who I felt had
showed the most interest” in my above brief contacts with them. However, I
have decided to remove specific names. Also see posted comments below.)
Ashish Sirohi
Paper "On the convergence
of certain infinite processes to rational numbers'' in PDF format
Email: as@infiniteseriestheorem.org ,
as7y@yahoo.com
Mathematical
Logic Issue. I have isolated the part of the paper that involves a
simple issue of mathematical logic as a self-contained and purely elementary
paper carrying the central theorem and its proof. You do not need to be
interested in or have any knowledge of any issues from Number Theory to read
this reduced version in PDF format and help decide an important
mathematics issue. Any written response explaining why the arguments
in my proof are fallacious and logically invalid is appreciated. Logical games
have been played with infinity before and been refuted (for example, Zeno's
paradox). I look forward to being shown my specific error, if any.
Possible points to ponder: (1) Irrational numbers literally are just names
given to non-recurring, non-terminating decimals (such as the name “pi”) --
is this not a mathematically valid statement (details in Part 7-2 of my paper)?
(2) If this is a mathematically valid way to differentiate between rational and
irrational numbers then can it not be used to derive and prove other differences
in properties? (3) If it can be so used, then exactly why and how is my use
wrong?
(Does the fact that not only are the statements elementary but so is the
proof create psychological issues in acknowledging that our mathematicians
missed the simple truths and were on the wrong track here?)
Click
here for some of the mathematical comments I have received from
mathematicians so far in response to this paper/website. (Last entry on July 10,
2003)*
*In
response to inquiries about status of the issues. I got weary of updating
comments, I have been busy. Activity continues, no final resolution. We are all
new to the internet, it was interesting to archive comments, but it is a matter
of free time and purpose.
Also, recent email discussions contain names of mathematicians, the name itself
being the interesting part; and as above I was keeping names out -- Ashish
Sirohi, Nov 6
03.
Update
Below is the final conclusion from three Fields Medallists I have corresponded
with.
One
of the three was very kind with his time and interest and we
exchanged many emails. He seems to agree that theorem 4 would be very useful and that theorem 2
could follow from it. However, he did not state whether he agrees that, assuming
theorem 4, I have proved that theorem 2 follows. He had many other comments on
many matters, and these were most useful.
The second said he has found "mistakes." Below is his email and my
reply. I tried to have him write further, and an influential colleague of his might have
also
urged him to do so. But it seems he will not be writing further.
His email:
After a careful reading of your paper my opinion
is that the statements are plausible but your arguments are fallacious and fail to give a
mathematical proof. If you cannot see this by yourself there is no point on my part to convince you of your mistakes.
It is my strict policy not to examine revisions and I will not consider further e-mail on the subject.
My
reply by email:
I do not even know where you found mistakes. Is
it in the proof of theorem 4/6 (with the two added conditions) or is it in how theorem 2
follows from these or is it in both? Theorem 2 follows naturally and in a standard mathematical manner from theorem 4/6, and if needed I can
revise and re-write any part of that proof to satisfy any objections. As for the proof of theorem 4/6 (with the two added conditions), I
would agree that Part 7-2 of Remark 7 does not look like mathematical proofs one generally reads, but that alone does not make it not a
mathematical proof.
I do not ask that you spend time trying to “to convince [me] of [my]
mistakes.” Can you please just tell me what the mistakes are? I have sent my paper to other mathematicians and it seems you are the only one
who is sure about what is wrong with my arguments. If needed, I will work with other mathematicians to understand what you have written and
will not write you saying I am not convinced.
I would be infinitely grateful if you could help resolve this
mathematical dilemma.
The third came to a different conclusion. We corresponded by email and had one
phone conversation. He stated that even if theorem 4 has been proven, nothing
such as theorem 2 could follow from it, because one could never be sure all
possibilities have been eliminated. He believes theorem 4 would therefore not be
usable and the entire method I am suggesting would never work.
His email:
The main problem is that just because the function f in theorem 4 is a
quotient of polynomials, this does not imply that the functions you break it
into in theorem 4 are also quotients of polynomials. I think it is unlikely that
this problem can be fixed with the elementary methods you are using. It is usually very hard to show that an infinite series has an irrational sum.
(Over the phone he emphasized his position that I am on the wrong track with
my approach; I insisted that important facts can be proved to follow from
theorem 4).
-- Ashish Sirohi, Nov 3 04
PDF files need Adobe Acrobat Reader. You can download
the free Adobe Acrobat Reader here
Some links, mostly from MathWorld, that give
references to papers and books on related subjects
This site was created mid-April 2003.
My other website, http://physicsnext.org
(new, April 2 2007). Also http://cuspeech.org
