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Fermat Numbers

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.

Pi (Π)

Pi, the mathematical constant, has a yet to determine value. As of now following sequence of digits has found to be most accurate for the approximation for $\pi$. :3.1415926535897932384626433832795028841971693993751058209 7494459230781640628620899862803482534211706798214808651 3282306647093844609550582231725359408128481117450284102 7019385211055596446229489549303819644288109756659334461 2847564823378678316527120190914564856692346034861045432 6648213393607260249141273724587006606315588174881520920 9628292540917153643678925903600113305305488204665213841 4695194151160943305727036575959195309218611738193261179 3105118548074462379962749567351885752724891227938183011 9491298336733624406566430860213949463952247371907021798 6094370277053921717629317675238467481846766940513200056 8127145263560827785771342757789609173637178721468440901 2249534301465495853710507922796892589235420199561121290 2196086403441815981362977477130996051870721134999999837 2978049951059731732816096318595024459455346908302642522 3082533446850352619311881710100031378387528865875332083 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381654729 : An Interesting Number Happened To Me Today

Image via Wikipedia

You might be thinking why am I writing about an individual number? Actually, in previous year annual exams, my registration number was 381654729. Which is just an ‘ordinary’ 9-digit long number. I never cared about it- and forgot it after exam results were announced. But today morning, when I opened “Mathematics Today” magazine’s October 2010, page 8; I was brilliantly shocked. 381654729 is a nine digit number with each of the digits from 1 to 9 appearing once. The whole number is divisible by 9. If you remove the right-most digit, the remaining eight-digit number is divisible by 8. Again removing the next-right-most digit leaves a seven-digit number that is divisible by 7. Similarly, removing next-rightmost digit leaves a six-digit number that is divisible by 6. This property continues all the way down to one digit.
Further research on this number provided a term for this number as Poly-divisible Number.
And I also noticed that a similar problem has been asked in U S A Mathematical Talent Search  competition. See the first question in the doc below:

To view this document in appropriate size click on View tab of the doc.

After this beautiful incident, I would like to quote a statement here:

Mathematical Wonders happen with Mathematicians.

Numbers always chase me.

Problem1: Smallest Autobiographical Number:

A number with ten digits or less is called autobiographical if its first digit (from the left) indicates the number of zeros it contains,the second digit the number of ones, third digit number of twos and so on.

For example: 42101000 is autobiographical.

Find, with explanation, the smallest autobiographical number.

Solution of Problem 1

Problem 2: Fit Rectangle:

A rectangle has dimensions $39.375$ cm $\times 136.5$ cm.

• Find the least number of squares that will fill the rectangle.
• Find the least number of squares that will fill the rectangle, if every square must be the same size and Find the largest square that can be tiled to completely fill the rectangle.

Solution of Problem 2

Solutions of Problem 1:

The restrictions which define an autobiographical number make it straightforward to find the lowest one. It cannot be 0, since by
definition the first digit must indicate the number of zeros in the number. Presumably then, the smallest possible autobiographical number will contain only one 0.If this is the case, then the first digit must be 1. 10 is not a candidate because the second digit must indicate the number of 1s in the number–in this case, 1. So If the
number contains only one zero, it must contain more than one 1.
(If it contained one 1 and one 0, then the first two digits
would be 11, which would be contradictory since it actually contains two 1s).
Again, presumably the lowest possible such number will contain the lowest
possible number of 1s, so we try a number with one 0 and two 1s. It will be of the form: 12-0–..
Now, there is one 2 in this number, so the first three digits must be 121. To meet all the conditions discussed above, we can simply take a 0 onto the end of this to obtain 1210, which is
the smallest auto-biographical number.

Solution of Problem 2:

We solve the second and third parts of the question
first:

We convert each number to a fraction and get a common denominator, then find the gcd (greatest common divisor) of the numerators.

That is, with side lengths $39.375$ cm and $136.5$ cm , we convert those numbers to fractions (with a common
denominator):
$39.375 = \dfrac{315}{8}$.

$136.5 = \dfrac{273}{2} = \dfrac{1092}{8}$.

Now we need to find the largest common factor of 315 and 1092.
Which is 21. So $\dfrac{21}{8}=2.625$ is the largest number that divides evenly into the two numbers $39.375$ and $136.5$.
There will be $\dfrac{1092}{21} \times \dfrac{315}{21} = 52 \times 15 = 780$ squares, each one a $2.625$ cm $\times 2.625$ cm square needed to fill the rectangle (52 in each row,with 15 rows).

Now we shall solve the first part.
Number of squares lengthwise is 52 and breadthwise is 15. Now we will combine these squares in order to find least number of squares to fill the rectangle. First three squares would be of
dimension 15 by 15. In this way length of 45 units is utilized. Now the rectangle which is left with us excluding three squares is 7 by 15. Again in the same way we can make two squares of dimension 7 by 7. In this way breadth of 14 units is utilized.
Now we are left with the rectangle of dimension 7 by 1.
These can further be subdivided into seven squares each of
dimension 1 by 1. In this way the least number of squares to fill the
rectangle is 3 + 2+ 7 = 12. The required answer is 12.
Note that the three numbers 3, 2, and 7 are involved in the Euclidean Algorithm for finding the g.c.d.!

Source: Internet

Free Online Algebra Books

Internet is full of knowledge. There are many professors who have shared their works online. I have listed a few books on Algebra and Related Mathematics in this article. I am not very serious, but I think these are very useful for Undergraduate and Graduate Students, including me.

1. Abstract Algebra OnLine by Prof. Beachy
2. Understanding Algebra by James Brennan
3. Abstract Algebra : Theory and Applications by Tom Judson
4. Elements of Abstract and Linear Algebra by E H Connell
5. Linear Algebra by Jim Hefferon
6. Elementary Linear Algebra by Keith Matthews
7. Linear Algebra, Infinite dimensions and Mapleby James Herod
8. Elementary Number Theory by William Stein
9. Abstract Algebra: The Basic Graduate Year, A Course in Algebraic Number Theory, and, A Course in Commutative Algebra are three ebooks by Robert Ash and are available here on his website.
10. A Course in Universal Algebra by Stanley Burris & H. P. Sankappanvar
11. An Introduction to the Theory of Numbers by Leo Moser
12. A Computational introduction to Number Theory and Algebra by Victor Shoup
13. Sets, Relations, Functions by Ivo Duentsch and Günther Gedigo
14. Group Theory by Pedrag Civitanovic
15. Linear & Multilinear Algebra by C C Wang & R M Bowen
16. Abelian Categories by Peter Freyd
17. Categories and Groupoids by P. J. Higgins
18. Lie Algebras by Prof. Sternberg

All books, except few are in Acrobat Portable Document Format. You may need acrobat pdf reader to read these. However, if you don’t have that- or hate pdf readers – try online pdf/ps reader at view.samurajdata.se.

This list is expandable. If you know any other book on Algebra which is available online for free, then please give a few seconds and put that into the Comment-Box below, with the link. (It supports HTML.)

A Problem on Infinite Sum and Recurrence Relations

Problem

Consider the infinite sum
$\mathbb{S} = \dfrac {a_0} {10^0} + \dfrac {a_1} {10^2} + \dfrac {a_2} {10^4} + .......$ where the sequence $\{a_n\}$ is defined by $a_0=a_1=1$ , and the recurrence relation $a_n=20a_{n-1} + 12 a_{n-2}$ for all positive integers $n \ge 2$. If $\sqrt {\mathbb {S} }$ can be expressed in the form $\dfrac {a} {\sqrt{b}}$ where $a$ & $b$ are relatively prime positive integers. Determine the ordered pair $(a, \, b)$ .

Solution

As the recurrence relation states $a_n-20a_{n-1} - 12 a_{n-2} =0$, we shall reform the infinite sum into the same pattern (have a deep look);
$\mathbb{S} - \dfrac {20 \mathbb{S} } {10^2} - \dfrac {12 \mathbb{S} } {10^4}$
$= ( \dfrac {a_0} {10^0} + \dfrac {a_1} {10^2} + \dfrac {a_2} {10^4} + \dfrac {a_3} {10^6} + .... )$ $- ( \dfrac {20a_0} {10^2} + \dfrac {20a_1} {10^4} + \dfrac {20a_2} {10^6} + \dfrac {20a_3} {10^8} + .... )$ $- ( \dfrac {12a_0} {10^4} + \dfrac {12a_1} {10^6} + \dfrac {12a_2} {10^8} + \dfrac {12a_3} {10^10} + .... )$

After Simplifying and arranging

$= \dfrac {a_0} {10^0} + \dfrac {a_1} {10^2} - \dfrac {20a_0} {10^2} + \dfrac {a_2 -20a_1-12a_0} {10^4} + \dfrac {a_3-20a_2-12a_1} {10^6} + \dfrac {a_4-20a_3-12a_2} {10^8} + . . . . . . \infty$
Now, as {the recurrence relation is}
$a_n-20a_{n-1}-12a_{n-2} =0$ for all $n \ge 2$, all terms except first three are zero in R.H.S.
Hence we have,
$\mathbb{S} - \dfrac {20 \mathbb{S} } {10^2} - \dfrac {12 \mathbb{S} } {10^4} = \dfrac {a_0} {10^0} + \dfrac {a_1} {10^2} - \dfrac {20a_0} {10^2}$
and substituting $a_0 =a_1=1$, we have
$\mathbb{S} - \dfrac {20 \mathbb{S} } {100} - \dfrac {12 \mathbb{S} } {10000}$
$= \dfrac {1} {1} + \dfrac {1} {100} - \dfrac {20} {100}$
or
$\dfrac {7988 \mathbb{S}} {10000} = \dfrac {81} {100}$
so,
$\mathbb{S} =2025/1997$
From the Problem,
$\sqrt {\mathbb{S}} = \sqrt{2025/1997} = 45/\sqrt{1997} =a/\sqrt{b}$
So, the desired ordered pair is $(a, b) = (45, 1997)$.

A Problem on Ordinary Nested Radicals

Problem

Prove or disprove that
$\sqrt {7+\sqrt{33}} + \sqrt {6+\sqrt{35} } = \sqrt {5+\sqrt{21}} + \sqrt {8+\sqrt{55}}$

Solution

In order to simplify the radicals, the radicands should be forced to equal square numbers (e.g., $7+\sqrt{33}$ should be a square of some number). Numbers whose squares have a rational and radical part are usually in the form $x+y$.
So let $\sqrt{7 +\sqrt{33}} = x+y = \sqrt{(x+y)^2} = \sqrt {x^2+y^2+2xy}$
and set
$x^2+y^2=7$ and $2xy=\sqrt{33} \, , i.e., y=\sqrt{33}/2x$
Thus $x^2 + \left ( \frac {\sqrt{33}}{2x} \right )^2 =7$
which on simplification yields $x=\sqrt{22}/2$
And also $y=\sqrt{6}/2$

Thus,
$\mathbf {\sqrt{7+\sqrt{33}} = \frac {\sqrt{22}+\sqrt{6}}{2} }$
Using the same process for other radicals:
$\mathbf {\sqrt{6+\sqrt{35}} = \frac{\sqrt{10}+\sqrt{14}}{2} }$
$\mathbf {\sqrt{8+\sqrt{55}} = \frac{\sqrt{10}+\sqrt{22}}{2} }$
$\mathbf {\sqrt{5+\sqrt{21}} = \frac{\sqrt{6}+\sqrt{14}}{2} }$
Thus, now we can easily prove (by addition) that

$\sqrt {7+\sqrt{33}} + \sqrt {6+\sqrt{35}} =\sqrt {5+\sqrt{21}} + \sqrt {8+\sqrt{55}}$