The paper sets forth simple methods to decide whether an infinite series or other infinite processes would converge to a rational number or an irrational number.
Theorem 4 of attached paper should be one of the fundamental theorems of mathematics or, if one makes an issue of mathematical logic and is not willing to accept the logic of the proof of Theorem 4, it should be a central conjecture of mathematics, assuming which the proof of astounding theorems such as Theorem 2 below follow indisputably and elegantly. Replacing addition with multiplication, we get parallel theorems for infinite products.
If some scientific truth is so elementary and easy that anyone could have seen it but nobody did -- is that good or bad? Let me choose an example to compare (and let us not argue about why this choice etc., it is just for illustration of a point). If you told physicists that (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 in the attached paper is slightly different from the one above).
Three examples of application of above thm: 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.)
Paper "On the convergence of certain infinite processes to rational numbers'' in PDF format
Email: as7y (at) yahoo (dot) com
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?)
*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.
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.
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.
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
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Some links, mostly from MathWorld, that give references to papers and books on related subjects
This site was created mid-April 2003.
My other sites:
Physics community's mistaken belief that Einstein's two 1905 postulates could only lead to one possible set of equations (except for one highly-respected physicist who broke from the crowd in mid-20th century and suggested otherwise).
Columbia University and Lee Bollinger's use the New York Police Department and U.S. Justice Department to prosecute freedom of speech