Painted Diary

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 extremely uncertain pattern at various number ranges. Some prime numbers may have difference of hundreds, while few others are as…

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 as the difference $n_2-n_1$, if $n_2 >n_1$. Moving on the proofs: PROOF OF PROPOSITION 1: Let $n_1 > n_2$, then $n_1=2n_2$.…

08 – 09-10 Of 11-12-13

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 setup permutations $ are used -- first of which being the usual day-month-year , second  month-day-year format and the third being…

Carleson

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 time on 3 June 2003. Following mathematicians have received this award since then:   Abel Prize, 2003 Jean Pierre Serre…

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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 the exam. The examinee needs to have high self-organization, ability to keep his emotions and absolute confidence in his abilities.…

Papers In Garbage

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 shown that many of the most famous revision techniques are pointless and shouldn’t be considered a guarantee for exam success.…

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. https://www.youtube.com/watch?v=YjuexKGHLTM 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 (x, y) -> (y - sign(x)*sqrt(abs(b*x - c)), a -x ) That is, $ x= y- \mathrm{sign}{(x)} \cdot \sqrt{…

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 fact that $ i^2=-1$ . Again, as $ \dfrac{i}{1}=\dfrac{1}{i}$ or, $ i^2=1$ or, $ i^2+2=1+2$ or, $ -1+2=3$ $ 1=3…

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 add them and get different results. Dmitri A. Borgmann, the German recreationalist, puzzler and father of logology, noticed the following expression…

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$ ...(iii) $ \dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{1}{d^2}+\dfrac{1}{e^2}=15$ ...(iv) $ abcde = -1 $ ...(v). Solve and find the values of a, b, c, d…

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 number theory. Euler told that the equation $ f(x)=x^2+x+k$ yields many prime numbers with the values of x being input…

Irrational Numbers Featured Image

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}$ is rational if and only if $ a$ and $ b$ are integers relatively prime to each other and $…