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Happy Holi! : The Village Tour

Friday, March 29th, 2013 15:46 / 1 Comment

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.

ENJOY READING! Err… Watching.

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Euler’s (Prime to) Prime Generating Equation

Tuesday, March 26th, 2013 09:35 / Leave a Comment

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

Tuesday, February 14th, 2012 19:18 / 3 Comments

“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. $x=\frac{a}{b} \in \mathbb{Q} \iff \mathrm{g.c.d.} (a,b) =1, \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus \{0\}$ .

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Gamma Function

Tuesday, February 7th, 2012 15:35 / Leave a Comment

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 $\Gamma (a) =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ .

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The Area of a Disk

Friday, January 27th, 2012 16:22 / 1 Comment

~~[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…)

10952186-3d-color-triangles-geometrical-forms-a-vector

Triangle Inequality

Friday, January 20th, 2012 10:57 / 5 Comments

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

Monday, January 9th, 2012 20:15 / 5 Comments

Cartoons and Comics are very useful in the process of explaining complicated topics, in a very light and humorous way. Like:
How Mathematicians Debate?
You're Godel, Boole etc. because you work with logic
(more…)

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MY DIGITAL NOTEBOOK

Category Archives: Math

Happy Holi! : The Village Tour

Euler’s (Prime to) Prime Generating Equation

Proofs of Irrationality

Gamma Function

The Area of a Disk

Triangle Inequality

My Five Favs in Math Webcomics

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