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Fermat Numbers

Wednesday, September 28th, 2011 06:00 / Leave a Comment

Fermat Number, a class of numbers, is an integer of the form $F_n=2^{2^n} +1 \ \ n \ge 0$ .

For example: Putting $n := 0,1,2 \ldots$ in $F_n=2^{2^n}$ we get $F_0=3$ , $F_1=5$ , $F_2=17$ , $F_3=257$ etc.

Fermat observed that all the integers $F_0, F_1, F_2, F_3, \ldots$ were prime numbers and announced that $F_n$ is a prime for each natural value of $n$ .

In writing to Prof. Mersenne, Fermat confidently announced:

I have found that numbers of the form $2^{2^n}+1$ are always prime numbers and have long since signified to analysts the truth of this theorem.

However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that $F_1 -F_4$ are primes but $F_5=2^{2^5} =4294967297$ is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that $F_5$ is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.

Theorem:

The Fermat number $F_5$ is divisible by $641$ i.e., $641|F_5$ .

Proof:

As defined $F_5 :=2^{2^5}+1=2^{32}+1 \ \ldots (1)$

Factorising $641$ in such a way that $641=640+1 =5 \times 128+1 \\ =5 \times 2^7 +1$
Assuming $a=5 \bigwedge b=2^7$ we have $ab+1=641$ .

Subtracting $a^4=5^4=625$ from 641, we get $ab+1-a^4=641-625=16=2^4 \ \ldots (2)$ .

Now again, equation (1) could be written as
$F_5=2^{32}+1 \\ \ =2^4 \times {(2^7)}^4+1 \\ \ =2^4 b^4 +1 \\ \ =(1+ab-a^4)b^4 +1 \\ \ =(1+ab)[a^4+(1-ab)(1+a^2b^2)] \\ \ =641 \times \mathrm{an \, Integer}$
Which gives that $641|F_n$ .

Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond $F_4$ . The best guess is that all Fermat numbers $F_n>F_4$ are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers $F_n, F_m \ m > n \ge 0$ , $\mathrm{gcd}(F_m, F_n) =1$ .

Following two theorems are very useful in determining the primality of Fermat numbers:

Pepin Test:

For $n \ge 1$ , the Fermat number $F_n$ is prime $\iff 3^{(F_n-1)/2} \equiv -1 \pmod {F_n}$

Euler- Lucas Theorem

Any prime divisor $p$ of $F_n$ , where $n \ge 2$ , is of form $p=k \cdot 2^{n+2}+1$ .

Fermat numbers ( $F_n$ ) with $n=0, 1, 2, 3, 4$ are prime; with $n=5,6,7,8,9,10,11$ have completely been factored; with $n=12, 13, 15, 16, 18, 19, 25, 27, 30$ have two or more prime factors known; with $n=17, 21, 23, 26, 28, 29, 31, 32$ have only one prime factor known; with $n=14,20,22,24$ have no factors known but proved composites. $F_{33}$ has not yet been proved either prime or composite.

Video: Documentary on Proof of Fermat’s Last Theorem (wpgaurav.wordpress.com)

Study Notes Announcements and more…

Thursday, September 22nd, 2011 19:12 / 2 Comments

Announcement

Hi all!
I know some friends, who don’t know what mathematics in real is, always blame me for the language of the blog. It is very complicated and detailed. I understand that it is. But MY DIGITAL NOTEBOOK is mainly prepared for my study and research on mathematical sciences. So, I don’t care about what people say (SAID) about the

A Torus

content and how many hits did my posts get. I feel happy in such a way that MY DIGITAL NOTEBOOK has satisfied me at its peak-est level. I would like to thank WordPress.com for their brilliant blogging tools and to my those friends, teachers and classmates who always encourage me about my passion. For me the most important thing is my study. More I learn, more I will go ahead. So, today (I mean tonight) I have decided to write some lecture-notes (say them study-notes, since I am not a lecturer) on MY DIGITAL NOTEBOOK. I have planned to write on Group Theory at first and then on Real Analysis. And this post is just to introduce you with some fundamental notations which will be used in those study-notes.

Notations

Conditionals and Operators
$r /; c$ : Relation $r$ holds under the condition $c$ .
$a=b$ : The expression $a$ is mathematically identical to $b$ .
$a \ne b$ : The expression a is mathematically different from $b$ .
$x > y$ : The quantity $x$ is greater than quantity $y$ .
$x \ge y$ : The quantity $x$ is greater than or equal to the quantity $y$ .
$x < y$ : The quantity $x$ is less than quantity $y$ .
$x \le y$ : The quantity $x$ is less than or equal to quantity $y$ .
$P := Q$ : Statement $P$ defines statement $Q$ .
$a \wedge b$ : a and b.
$a \vee b$ : a or b.
$\forall a$ : for all $a$ .
$\exists$ : [there] exists.
$\iff$ : If and only if.
Sets & Domains
$\{ a_1, a_2, \ldots, a_n \}$ : A finite set with some elements $a_1, a_2, \ldots, a_n$ .
$\{ a_1, a_2, \ldots, a_n \ldots \}$ : An infinite set with elements $a_1, a_2, \ldots$
$\mathrm{\{ listElement /; domainSpecification\}}$ : A sequence of elements `listElement` with some `domainSpecifications` in the set. For example, $\{ x : x=\frac{p}{q} /; p \in \mathbb{Z}, q \in \mathbb{N^+}\}$ $a \in A$ : $a$ is an element of the set A.
$a \notin A$ : a is not an element of the set A.
$x \in (a,b)$ : The number x lies within the specified interval $(a,b)$ .
$x \notin (a,b)$ : The number x does not belong to the specified interval $(a,b)$ . Standard Set Notations
$\mathbb{N}$ : the set of natural numbers $\{0, 1, 2, \ldots \}$
$\mathbb{N}^+$ : The set of positive natural numbers: $\{1, 2, 3, \ldots \}$
$\mathbb{Z}$ : The set of integers $\{ 0, \pm 1, \pm 2, \ldots\}$
$\mathbb{Q}$ : The set of rational numbers
$\mathbb{R}$ : The set of real numbers
$\mathbb{C}$ : The set of complex numbers
$\mathbb{P}$ : The set of prime numbers.
$\{ \}$ : The empty set.
$\{ A \otimes B \}$ : The ordered set of sets $A$ and $B$ .
$n!$ : Factorial of n: $n!=1\cdot 2 \cdot 3 \ldots (n-1) n /; n \in \mathbb{N}$

Other mathematical notations, constants and terms will be introduced as their need.

For Non-Mathematicians:
Don’t worry I have planned to post more fun. Let’s see how the time proceeds!

Using math symbols in wordpress pages and posts – LaTex (1) (etidhor.wordpress.com)
Do complex numbers really exist? (math.stackexchange.com)
LaTeX Typesetting – Basic Mathematics (r-bloggers.com)

Social Networks for Math Majors

Tuesday, September 13th, 2011 11:44 / 9 Comments

Math or Mathematics is not as difficult as it is thought to be. Mathematical Patterns, Structures, Geometry and its use in everyday life make it beautiful. ‘Math majors’ term generally include Math students, Math professors and researchers or Mathematicians. Internet has always been a tonic for learners and whole internet is supposed to be a social network, in which one shares and others read, one asks & others answer. There are thousands of social networks (and growing) where you enjoy your days, share fun etc. However there are only a few social (mathematical) networks which are completely focused on math and related sciences. But these are brilliantly good enough to demonstrate the wisdom of mathematicians. I have tried to list my favorite social networking websites on mathematics. Please have a read and give feedback in form of comments
Click On Images To Visit Corresponding Websites.

Math.Stack Exchange

Mathematics StackExchange Website

Mathematics Stack exchange is a website dedicated to all types of mathematical discussions. You can ask questions, give answers, comment on questions and vote for it. Registration is very easy and takes seconds. Depending on your work, you are given ‘reputations’. Depending upon some special works, you are also given some privileges.

This is a free, community driven Q&A for people studying math at any level and professionals in related fields. It is a part of the Stack Exchange network of Q&A websites, and it was created through the open democratic process defined at Stack Exchange Area 51. (more…)

Nanostory of Nanotechnology

Sunday, August 21st, 2011 19:59 / 2 Comments

Fullerene

Well, this is not going to be a nano [very short] story either of fairies or aliens. This is a big story of Nanotechnology, one of the most advanced topics in physics. Wait. It’s not going to be so hard or advanced to read. It is really going to be a good story because I’m not going to teach you about this stuff. I am trying to say and save it’s history on MY DIGITAL NOTEBOOK. I think you all should also read this.
Nanotechnology has become a widely discussed topic today in newspapers, magazines, journals, blogs and even in television ads. It’s very common that some organization announcing ‘yet another’ “nano-conference”. Nanotechnology or nano tech in short, refers to the technology of creating materials, devices and functions using atomically manipulated matter. If you didn’t understand it clearly yet read further lines. (more…)

NPTEL: An Innovation in Visual and Online Learning

Sunday, August 14th, 2011 21:20 / 3 Comments

First of all, Happy Independence Day to all my Indian Friends and followers. This post is about an Indian pioneer in online learning, namely NPTEL.

National Programm on Technology Enhanced Learning (NPTEL) provides E-learning through online Web and Video courses in Engineering, Science and humanities streams. The mission of NPTEL is to enhance the quality of Engineering education in the country by providing free online courseware. All videos of NPTEL include the lectures of Indian professors in IITs and IISc and they can be found either on NPTEL homepage or at their YouTube page. On Youtube, the Video Courses are organised as PLAYLISTS under the following Categories:

Core Sciences
Civil Engineering
Computer Science and Engineering
Electrical Engineering
Electronics and Communication Engineering
Mechanical Engineering

.
And on their website, these are arranged in a order of Subjects.
$\textnormal {Visit NPTEL Website}$
$\textnormal{Watch Videos On YouTube.com}$
Bellow are two Course Videos, as a demo, one on Semiconductors and other on Artificial Intelligence. It would be better to watch them, before you go for whole.
(more…)

Blog of the Month -August 2011

Friday, August 5th, 2011 20:40 / 7 Comments

I announced that I shall chose a blog from the education blogsphere as Blog of the Month. To complete this task, I googled for days, read them, analysed them and now I have the winner of ‘Blog of the Month’.
This is the first month of this series and discussing article is made in hurry, so one can feel an emptiness and lack of interest in it. But believe, Blog of the monthwas not selected in hurry. I took quick looks on about 500 blogs and thousands of posts. I created a list of all blogs I read and rated them on behalf of their qualities, visitors, content, language etc. From the list of 513 blogs, the shortlisted blogs were:

What’s New (math)
Gödel’s Lost Letter and P=NP(Math and Computer Science)
Peter Cameron’s Blog(math)
Let’s Play Math(math)
Unapologetic Mathematician(math)
Cock Tail Party Physics(Physics)
WordPress Tips(Blogging)
Honglang Wang’s Blog (Math and Programming)
The GeomBlog(CS)
Republic Of Mathematics (Math and Media)

I count a lot of things that there’s no need to count. Just because that’s the way I am. But I count all the things that need to be counted.

And Yes! The blog of the month is Peter Cameron’s Blog with useful content, interactive language and multidimensional approach to mathematics.

About Peter Cameron’s Blog

Peter Cameron is a professor of mathematics in London and he writes about math, media and education at http://cameroncounts.wordpress.com. He mingles everything with math, like poetry – media – fun and internet. His blog is full of Expositories, Problems and Results, Posts about doing – playing and learning mathematics, Poetry, Events Talks and Conferences, Typesettings and Mathematics in Media. A list of categorized posts can be found here.

Reviews

Rating: 8.9/10
View: 7.0/10
Content: 9.5/10
Interaction: 9.0/10
Language: 9.5/10
Frequency of Posts: 8.5/10
Content Management: 10/10

[Last Updated: 20:03 IST 2011/08/05]

What are you views and thoughts on this selection? Rate Peter Cameron’s blog on the base of 10. Your comments are heartly welcomed.

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An Important Announcement

Sunday, July 31st, 2011 18:18 / 3 Comments

I have decided to change the style of posts and the posting frequency. Worry not! I am not leaving math, physics and chemistry. I am now an experienced campaigner and I have just arranged the things as follow.

I ‘will’ post atleast 3 posts a week.
Atleast one post will be a ‘problem’ post in which problem solving skills are tested. Another one of the posts would be about applications of mathematical sciences.
On 5th day of every month, I shall chose a blog from World Wide Web and write a review on it.
Other things that I would try in my posts are:
- Mathematical Puzzles
- Valuable Images
- Success Stories of Mathematicians / Physicists / Chemists and students.
- Learning Techniques

I have thought a lot about this blog. This is going to be a ‘wow’ in education blogging. God help me.

Gaurav Tiwari

Please let me know if you have any suggestions about the structure of posts and writing style.

MY DIGITAL NOTEBOOK

Tag Archives: science

Fermat Numbers

Theorem:

Proof:

Pepin Test:

Euler- Lucas Theorem

Related articles

Study Notes Announcements and more…

Announcement

Notations

Nanostory of Nanotechnology

NPTEL: An Innovation in Visual and Online Learning

Blog of the Month -August 2011

About Peter Cameron’s Blog

Reviews

An Important Announcement

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Tag Archives: science

Theorem:

Proof:

Pepin Test:

Euler- Lucas Theorem

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Announcement

Notations

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Math.Stack Exchange

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