Painted Diary
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This Prime Generating Product generates successive prime factors

Any integer greater than 1 is called a prime number if and only if its positive factors are 1 and the number p itself. The basic ideology involved in this post is flawed and the post has now been moved to Archives. – The Editor Prime Generating Formulas We all know how hard it is to predict a formula for prime numbers! They have…

Difference Paradox

Consider two natural numbers $n_1$ and $n_2$, out of which one is twice as large as the other. We are not told whether $n_1$ is larger or $n_2$, we can state following two propositions: PROPOSITION 1: The difference $n_1-n_2$, if $n_1 >n_2$, is different from the difference $n_2-n_1$, if $n_2 >n_1$. PROPOSITION 2: The difference $n_1-n_2$, if $n_1 >n_2$, is the same…
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11-12-13 – Last such sequence of the century

11th December 2013 (or in short 11-12-13) is just a few hours away here. It’s the last date of the 21st century with  such an extraordinary pattern of numbers.  In any calendar of the world, such date will be seen in 22nd century after 87 years and 54 days from today — on February 3, 2101, if all kind of date…
Carleson
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Abel Prize Laureates

Abel prize is one of the most prestigious awards given for outstanding contribution in mathematics, often considered as the Nobel Prize of Mathematics. Niels Henrik Abel Memorial fund, established on 1 January 2002, awards the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 1010000 USD) and was awarded for the first…
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Examination Strategies : Tactics & Tips

Every student or graduate knows how hard the first experience of passing exams is. Preliminary preparation starves the nervous system and the physical condition of the human body, however, the exam itself is always a stressful situation, which requires a candidate a great manifestation of mental and physical abilities. Therefore, just the knowledge of a subject is not enough for…
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Some good, OK, and useless revision techniques

Exam have been haunting student since forever, and although you’re willing to do whatever you can to retain essential information, sometimes you end up spending weeks studying with useless revision techniques. We’re accustomed to employ our own techniques when it comes to studying such as making sticky notes, highlighting, or drawing charts. However, recent studies conducted in the US have…

Hopalong Orbits Visualizer: Stunning WebGL Experiment

Just discovered Barry Martin’s Hopalong Orbits Visualizer — an excellent abstract visualization, which is rendered in 3D using Hopalong Attractor algorithm, WebGL and Mrdoob’s three.js project. Hop to the source website using your desktop browser (with WebGl and Javascript support) and enjoy the magic. PS: Hopalong Attractor Algorithm Hopalong Attractor predicts the locus of points in 2D using this algorithm…

Smart Fallacies: i=1, 1= 2 and 1= 3

This mathematical fallacy is due to a simple assumption, that $ -1=\dfrac{-1}{1}=\dfrac{1}{-1}$ . Proceeding with $ \dfrac{-1}{1}=\dfrac{1}{-1}$ and taking square-roots of both sides, we get: $ \dfrac{\sqrt{-1}}{\sqrt{1}}=\dfrac{\sqrt{1}}{\sqrt{-1}}$ Now, as the Euler’s constant $ i= \sqrt{-1}$ and $ \sqrt{1}=1$ , we can have $ \dfrac{i}{1}=\dfrac{1}{i} \ldots \{1 \}$ $ \Rightarrow i^2=1 \ldots \{2 \}$ . This is complete contradiction to the…

Four way valid expression

People really like to twist the numbers and digits bringing fun into life. For example, someone asks, “how much is two and two?” : the answer should be four according to basic (decimal based) arithmetic. But the same  with base three (in ternary number system) equals to 11. Two and Two also equals to Twenty Two. Similarly there are many ways you can…

Interesting Egyptian Fraction Problem

Here is an interesting mathematical puzzle alike problem involving the use of Egyptian fractions, whose solution sufficiently uses the basic algebra. Problem Let a, b, c, d and e be five non-zero complex numbers, and; $ a + b + c + d + e = -1$ … (i) $ a^2+b^2+c^2+d^2+e^2=15$ …(ii) $ \dfrac{1}{a} + \dfrac{1}{b} +\dfrac{1}{c} +\dfrac{1}{d} +\dfrac{1}{e}= -1$…

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 but still stun the mathematicians. The prime generating equation by Euler is a very specific binomial equation on prime numbers and yields more primes than any other relations out there in…
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Irrational Numbers and The Proofs of their 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 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}$…
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The Area of a Disk

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…
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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 $…

On Ramanujan’s Nested Radicals

Ramanujan (1887-1920) discovered some formulas on algebraic nested radicals. This article is based on one of those formulas. The main aim of this article is to discuss and derive them intuitively. Nested radicals have many applications in Number Theory as well as in Numerical Methods . The simple binomial theorem of degree 2 can be written as: $ {(x+a)}^2=x^2+2xa+a^2 \…
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Calendar Formula: Finding the Week-days

This is the last month of the glorious prime year 2011. We are all set to welcome upcoming 2012, which is not a prime but a leap year. Calendars have very decent stories and since this blog is based on mathematical approach, let we talk about the mathematical aspects of calendars. The international calendar we use is called Gregorian Calendar,…

How Many Fishes in One Year? [A Puzzle in Making]

This is a puzzle which I told to my classmates during a talk, a few days before. I did not represent it as a puzzle, but a talk suggesting the importance of Math in general life. This is partially solved for me and I hope you will run your brain-horse to help me solve it completely. If you didn’t notice,…
yes no puzzle
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A Yes No Puzzle

This is not just math, but a very good test for linguistic reasoning. If you are serious about this test and think that you’ve a sharp [at least average] brain then read the statement (only) below –summarize it –find the conclusion and then answer that whether summary of the statement is Yes or No. [And if you’re not serious about…