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## Happy Holi! : The Village Tour

Holi, the festival of colors, was celebrated this year on 27th and 28th of March all over India. I decided to move to my own village, Kasturwa and then to Surya’s Village, Shiv Patti, on this occasion. Here are some images from the events  taken with my Nokia device, which I thought were worth sharing.

## Euler’s (Prime to) Prime Generating Equation

The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent. The prime generating equation by Euler is a binomial which is actually very specific and yields more primes than any other relations out there in number theory. Euler told that the equation $f(x)=x^2+x+k$ yields many prime numbers with the values of x being input from x=0 to x=k-2; k being a prime.

Let’s see how many primes we can get by using different values of k and x:

 Serial Number Value of k (prime) Value of x (from x=0 to x=k-2) Value of f(x)=(x^2+x)+k Not a Prime? 1 2 0 2 2 3 0 3 3 1 5 4 5 0 5 5 1 7 6 2 11 7 3 17 8 7 0 7 9 1 9 No 10 2 13 11 3 19 12 4 27 No 13 5 37 14 11 0 11 15 1 13 16 2 17 17 3 23 18 4 31 19 5 41 20 6 53 21 7 67 22 8 83 23 9 101 24 13 0 13 25 1 15 No 26 2 19 27 3 25 No 28 4 33 No 29 5 43 30 6 55 No 31 7 69 No 32 8 85 NO 33 9 103 34 10 123 No 35 11 145 No 36 17 0 17 37 1 19 38 2 23 39 3 29 40 4 37 41 5 47 42 6 59 43 7 73 44 8 89 45 9 107 46 10 127 47 11 149 48 12 173 49 13 199 50 14 227 51 15 257 52 19 0 19 53 1 21 54 2 25 No 55 3 31 56 4 39 No 57 5 49 58 6 61 59 7 75 No 60 8 91 No 61 9 109 62 10 129 No 63 11 151 64 12 175 No 65 13 201 No 66 14 229 67 15 259 68 16 291 69 17 325 No 70 23 0 23 71 1 25 No 72 2 29 73 3 35 No 74 4 43 75 5 53 76 6 65 No 77 7 79 78 8 95 No 79 9 113 80 10. 133 81 11 155 No 82 12 179 83 13 205 No 84 14 233 85 15 263 86 87 16 295 No 88 17 329 89 18 365 No 90 19 403 91 20 443 92 21 485 No

The above table yields many prime numbers, which again can be put at the place of k and so on the table can be progressed.

According to Euler, 41 was the most appropriate value of k yielding more prime numbers than any other k. In the list below, each value of f(x) is a prime for k=41:

 k 41 x 0 f(x) 41 1 43 2 47 3 53 4 61 5 71 6 83 7 97 8 113 9 131 10 151 11 173 12 197 13 223 14 251 15 281 16 313 17 347 18 383 19 421 20 461 21 503 22 547 23 593 24 641 25 691 26 743 27 797 28 853 29 911 30 971 31 1033 32 1097 33 1163 34 1231 35 1301 36 1373 37 1447 38 1523 39 1601

So, the Euler’s Prime Generating Equation can be written as
$f(x) = x^2+x+41$ ; where x is an integer ranging from 0 to 39.

Wait. What if we increase the value of x beyond the limit of 39? What will we get?

The next values of f(x) in this series would be 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393, … .
Are all these prime numbers too? The answer is no. 1681 is not a prime number, neither are 1763 and 2021. Though all others are prime numbers.

## Proofs of Irrationality

“Irrational numbers are those real numbers which are not rational numbers!”

Def.1: Rational Number

A rational number is a real number which can be expressed in the form of $\frac{a}{b}$ where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero.
Following two statements are equivalent to the definition 1.
1. $x=\frac{a}{b}$ is rational if and only if $a$ and $b$ are integers relatively prime to each other and $b$ does not equal to zero.
2. $x=\frac{a}{b} \in \mathbb{Q} \iff \mathrm{g.c.d.} (a,b) =1, \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus \{0\}$.

## Gamma Function

If we consider the integral $I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ , it is once seen to be an infinite and improper integral. This integral is infinite because the upper limit of integration is infinite and it is improper because $t=0$ is a point of infinite discontinuity of the integrand, if $a<1$, where $a$ is either real number or real part of a complex number. This integral is known as Euler’s Integral. This is of a great importance in mathematical analysis and calculus. The result, i.e., integral, is defined as a new function of real number $a$, as $\Gamma (a) =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ .

## The Area of a Disk

[This post is under review.]

If you are aware of elementary facts of geometry, then you might know that the area of a disk with radius $R$ is $\pi R^2$.

The radius is actually the measure(length) of a line joining the center of disk and any point on the circumference of the disk or any other circular lamina. Radius for a disk is always same, irrespective of the location of point at circumference to which you are joining the center of disk. The area of disk is defined as the ‘measure of surface‘ surrounded by the round edge (circumference) of the disk.

The area of a disk can be derived by breaking it into a number of identical parts of disk as units — calculating their areas and summing them up till disk is reformed. There are many ways to imagine a unit of disk. We can imagine the disk to be made up of several concentric very thin rings increasing in radius from zero to the radius of disc. In this method we can take an arbitrary ring, calculate its area and then in similar manner, induce areas of other rings -sum them till whole disk is obtained. (more…)

## Triangle Inequality

Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If $a$, $b$ and $c$ be the three sides of a triangle, then neither $a$ can be greater than $b+c$, nor$b$ can be greater than $c+a$, nor $c$ can be than $a+b$.

Triangle

Consider the triangle in the image, side $a$ shall be equal to the sum of other two sides $b$ and $c$, only if the triangle behaves like a straight line. Thinking practically, one can say that one side is formed by joining the end points of two other sides.
In modulus form, $|x+y|$ represents the side $a$ if $|x|$ represents side $b$ and $|y|$ represents side $c$. A modulus is nothing, but the distance of a point on the number line from point zero.

Visual representation of Triangle inequality

For example, the distance of $5$ and $-5$ from $0$ on the initial line is $5$. So we may write that $|5|=|-5|=5$.

Triangle inequalities are not only valid for real numbers but also for complex numbers, vectors and in Euclidean spaces. In this article, I shall discuss them separately. (more…)

## My Five Favs in Math Webcomics

Cartoons and Comics are very useful in the process of explaining complicated topics, in a very light and humorous way. Like:

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