Popular-Lonely primes understood

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While reading standup mathematician Matt Parker‘s book Things to Make and do in Fourth Dimension, I found answer (on pp. 146) to the question I raised 7 months ago.

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When the grid happens to be a multiple of 6 wide, suddenly all primes snap into dead-straight lines. All primes (except 2 and 3) are one more or less than a multiple of 6. (© Matt Parker, 2014)

He also proves the following surprising theorem:

The square of every prime number greater than 3 is one more than a multiple of 24.

Let p be an odd prime not equal to 3. Now we subtract one from the square of this prime number. Therefore, we wish to prove that p^2-1=(p-1)(p+1) is a multiple of 24.

Note that, p^2-1 is a product of two even numbers. In particular, one of these two even numbers must be a multiple of 4, as they are consecutive even numbers and every other even number is divisible by 4. Hence we conclude that p^2-1 is divisible by 8. 

Observe that exactly one of three consecutive numbers, p-1,p,p+1 must be divisible by 3. Since p is an odd prime different from 3, one of p-1 or p+1 must be divisible by 3. Hence we conclude that p^2-1 is divisible by 3.

Combining both the conclusions made above, we complete proof of our statement (since 2 and 3 are coprime).

 

Special Numbers: update

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This post is a continuation of my earlier post: Special Numbers

Four (4)

This is the only euclidean space with properties different from other n-dimensional euclidean spaces. For example, there are smooth 4-manifolds which are homeomorphic but not diffeomorphic.  Put differently, for any dimension except four there is only one differentiable structure on the space underlying the Euclidean space of that dimension. For a discussion in this direction see this article by Liviu Nicolaescu. For other special properties of 4-dimesnions read Wikipedia article on 4-manifold.
Thanks to Dr. Ritwik Mukherjee for explaining this fact about four-space.

Hyperbolic Plane Example

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Few months ago I gave a lecture on Non-euclidean geometry and it was a bit difficult for me to give audience an example of hyperbolic surface from their day-to-day life. While reading Donal O’ Shea’s book on Poincaré Conjecture I came across following interesting example on pp. 97 :

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Negatively curved cloth will drape a woman’s side (© Donal O’ Shea, 2007)

Estrogen causes fat to be stored in the buttocks, thighs, and hips in women. Thus females generally have relatively narrow waists and large buttocks, and this along with wide hips make for a wider hip section and a lower waist-hip ratio compared to men. The saddle-shaped area on a woman’s side above her hip has negative curvature.

One can imagine cloth (it is flexible but does not stretch, hence an isometry) that would drape it perfectly. Here the region inside a circle of given radius contains more material than the same circle on the plane, and to make the cloth the tailor might start with a flat piece of fabric, make a cut as if he/she were going to make a dart, but instead of stitching the cut edges together, insert an extra piece of fabric or a gusset. Negatively curved cloth would have lots of folds if one tried to lay it flat in  dresser.

If one tries to extend a cloth with constant positive curvature (like a cap), in all directions, it would close up, making a sphere. On the other hand, if one imagines extending a cloth with constant negative curvature in all directions, one gets a surface called hyperbolic plane.

Opinions Welcome : Should we dream?

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We are living organisms and our biological objective of life is to grow and reproduce. Now if we add the social prospective to this basic concept of living organisms, we must ensure healthy life of our offspring (love, education, health, money…). So, a life without dream would be to work hard enough to be a able to use your capabilities to earn enough money to support your family.

Now, that was very idealistic view of life. Most parents “dream” that their children should achieve much more than them in their lives (like the objective of evolution). But surely “dreams” like to become richest person, greatest scientist etc. are not necessary for living. Moreover many times we have to pay huge price for chasing our dreams like less time for family (which is actually the reason for our existence), more stress, less friends, etc. I think that “dream chasing” is like drug addiction, it becomes inseparable part of your life and leads to various negative effects. So, I would like to ask:

Should we dream?

Maybe, working hard enough to fully utilise our capabilities is what dreaming all about.

Similar posts in past: 

Why I want to be a Mathematician?

Inquisitive Mathematical Thinking

Bitter Truth of Love

Solution to the decimal problem

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In this post I will discuss the solution of the problem I posted a week ago. Firstly I would thank Prof. Purusottam Rath for pointing out that this problem has already been solved. In 1961, Kurt Mahler published the solution in Lectures on Diophantine Approximations. In fact, M=0.12345678.... is called Mahler’s number since Mahler  showed it to be transcendental. For complete proof refer, Section 1.6 of “Making transcendence transparent: an intuitive approach to classical transcendental number theory“, by Edward Burger and Robert Tubbs, Springer-Verlag (2004). But I can give an outline of proof here.

The most basic type of transcendental numbers are the Liouville’s Numbers, these numbers satisfy following theorem (proved here on pp. 19):

Liouville’s Theorem: Let \alpha be a real number . Suppose there exists an infinite sequence of rational numbers p_n/q_n satisfying the inequality \displaystyle{\left|\alpha - \frac{p_n}{q_n}\right|<\frac{1}{q_n^n}}. Then \alpha is transcendental.

To be able to apply this theorem we use truncation procedure i.e. obtain the approximations of the number \alpha  by truncating the decimal expansion of \alpha immediately before each long run of zeros, and using this to get the desired inequality.

For Mahler number, Liouville’s theorem alone is not sufficient. Since, if we attempt truncation procedure, we will see that the number of decimal digits before each run of zeros far exceeds the length of the run. For example, a run of 2 zeros occurs after 189 digits. But, using Liouville’s theorem we can prove a partial result:

(1) There exists an infinite sequence of rational numbers p_n/q_n satisfying the inequality \displaystyle{\left|M - \frac{p_n}{q_n}\right|<\frac{1}{q_n^{4.5}}}. Hence, Mahler number M is either a transcendental number or an algebraic number of degree at least 5.

Since we need a stronger inequality, we will use following theorem (proved here on pp. 54), which states that:

Thue-Seigel-Roth Theorem: Let \alpha be an irrational algebraic number. Then for any \varepsilon > 0 there exists a constant c(\alpha, \varepsilon) depending on \alpha  such that \displaystyle{\frac{c(\alpha,\varepsilon)}{q^{2+\varepsilon}}<\left|\alpha - \frac{p}{q}\right|}

From this theorem we conclude that

(2) If M is algebraic of degree d\geq 2 (I showed in previous post that it is irrational) then for \varepsilon =0.5 there exists a constant c such that for all p_n/q_n, \displaystyle{\frac{c}{q_n^{2.5}}<\left|M-\frac{p_n}{q_n}\right|}

Using (1) and (2) we conclude that for all n,

\displaystyle{0<c<\frac{1}{q_n^2}}

But as q_n\rightarrow \infty, this inequality cannot hold. Hence M is transcendental.

This number, 0.123456789101112... is also known as Champernowne’s constant. I shall discuss more about it in future posts.

 

A Decimal Problem

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I would like to share a jotting from my diary (dated: 21-April-2016) which is bothering me:

Is the (decimal) number generated the concatenating all natural numbers like 0.1234567891011…, a transcendental number?

We can see that this number is irrational. Since, if it is a rational number then there exists a natural number n such that 0.\overline{123...n}=0.123...n..., which is clearly impossible.

Now, to prove a number is transcendental or not is a difficult question to answer (some open problems). So, either produce a polynomial equation which has this number as a solution or prove that no such polynomial equation exists.

Mathematical Relations

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In this post I will share my perception of relation of mathematics with other academic disciplines. All this is based on my very limited knowledge of various disciplines.

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Shape doesn’t signify anything.

Mathematics deals with study of properties of numbers (or the symbols representing them) and geometric objects (not in classical sense, it can mean manifolds also). In my opinion, there is no partition of mathematics into “applied” or “pure”, but intersections with other subjects. The term applied Mathematics doesn’t make any sense to me. Mathematics is somehow applicable in various places. For me, mathematics is what people call “pure” mathematics (what about “impure” Mathematics??).  Also now I agree with the vastly established belief that art and mathematics are similar, since both involve abstract ideas motivated but physical situations (at some point).

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Truth Lies Deception and Coverups – Democracy Under Fire (Source: http://goo.gl/yUHi93)

All experimental sciences (physics, chemistry, biology, economics) are based on statistics. Since statistics is a young discipline (only a couple of centuries old) many times we get wrong interpretation of results. As far as real life is concerned, study of statistics gives us a powerful tool for predicting future and Probability Theory acts as the connecting link between statistics and mathematics. Understanding of statistics affects us on daily basis since (effective) government policies are framed keeping statistical analysis in mind. Unfortunately, most of universities don’t have separate department for statistics.

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P vs NP Problem in Relationships (http://ctp200.com/comic/6; CC BY-NC 4.0)

Study of algorithms is one of the most important aspect of computer science (I am not talking about software industry…). What surprises me is that Euclid’s division algorithm is  one of the most efficient division algorithm even for computers! The neglected subject of Logic, which is supposed to be foundations of mathematics, flourishes in computer science. P vs NP is another “millennium open problem“.

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Convincing (http://xkcd.com/833/ ; CC BY-NC 2.5)

For me, Economics like Statistics is full of imperfections due to real life complications (so many dependencies to account for). Game Theory appears to be the connecting link between mathematics and economics.

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We all know that the needs of physicists are responsible for development of calculus and study of differential equations. On the other hand, theoretical physics (quantum mechanics, string theory) depends heavily on the developments in algebra.