### - Ashish Sirohi

My below main math theorem is discussed briefly in chapters 9 and 12 of my book, Toppling Relativity: My Struggle With the Church of Physics and Other Evaders of Truth, published Dec 2022. Book is Available at Amazon. Short Chapter excerpts are available at ashishsirohi.com. However, the book is largely about other matters. The detailed math theorem related discussion here and detailed PDFs are not in the book.

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 states that for an infinite series to converge to a rational number it must contain a hidden pattern whereby the infinite series collapses to a few finite terms, with the rest of the terms necessarily canceling each other as the series proceeds to infinity. If such a cancellation pattern does not exist then the infinite series would necessarily converge to an irrational number. The theorem implies that almost all (which is a mathematics term) infinite series would go to irrational numbers. Since all numbers could be represented by infinite series, this means that the infinite set of irrational numbers is larger than the infinite set of rational numbers.

From Theorem 4 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.

Theorem 2: Left $$f\left( n \right)$$ be a function which is either always positive or always negativefor all integers $$n~>~N$$, where $$N$$ is a positive integer. Further let $$f\left( n \right)=\frac{{p\left( n \right)}}{{q\left( n \right)}}$$, where $$p$$ and $$q$$ are polynomials of $$n$$ with rational coefficients and let $$\displaystyle {\sum }_{{\mathrm{n}} = 1}^{\infty}$$ $$\displaystyle ~f(n)$$ be convergent series. Then the series converges to a rational number if and only if f(n) can be broken into partial fractions $${{f}_{1}}\left( n \right),~~.~.~.~,~{{f}_{j}}\left( n \right)$$ such that $${{f}_{1}}\left( {n+{{i}_{1}}} \right)+.~.~.+~{{f}_{j}}\left( {n+{{i}_{j}}} \right)=0$$ for some integers $$~{{i}_{1}},~.~.~.,~{{i}_{j}}$$.
(The actual Theorem $$2$$ in the attached paper is slightly different from the one above).
Three examples of application of above thm: $$f\left( n \right)=\frac{1}{{\left( {1+6n} \right)\left( {5+6n} \right)}}$$ will give an infinite series converging to an irrational number whereas $$f\left( n \right)=\frac{1}{{\left( {1+6n} \right)\left( {7+6n} \right)}}$$ will give one going to an rational number. Also $$f\left( n \right)=\frac{1}{{{{n}^{k}}}}~$$, $$k$$ is an integer and $$k>1$$ will give an infinite series that converges to an irrational number.

Ashish Sirohi

Paper On the convergence of certain infinite processes to rational numbers in PDF format

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 ponder an important mathematics issue.

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?

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)

Below is the conclusion from three Fields Medallists I had 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.

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

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:

ashishsirohi.com

churchofphysics.org

cuspeech.org

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