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…

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A Possible Proof of Collatz Conjecture

Our reader Eswar Chellappa has sent his work on the solution of '3X+1' problem, also called Collatz Conjecture. He had been working on the proof of Collatz Conjecture off and on for almost ten years. The Collatz Conjecture can be quoted as follow: Let $\phi : \mathbb{N} \to \mathbb{N}^+$ be a function defined  such that: $$\phi(x):= \begin{cases} \frac{x}{2}, & \text{if } x \text{ is even } \\ 3x+1, & \text{ if } x \text{ is odd} \end{cases}$$ Then the iterates of…

Mobile Number Is Prime Number

My mobile number is a prime number

My personal mobile number 9565804301 is a prime number. What is a prime number? Any integer p greater than 1 is called a prime number if and only if its positive factors are 1 and the number p itself. In other words, the natural numbers which are completely divisible by 1 and themselves only and have no other factors, are called prime numbers. 2 ,3 ,5 ,7 ,11 ,13 ,17 , 19 ,23 ,29 …  etc. are prime numbers [or just Primes]. The numbers greater than 1, which are not prime are called Composite numbers.  …

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 $n_1=2n_2$.…

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Complete Elementary Analysis of Nested radicals

This is a continuation of the series of summer projects sponsored by department of science and technology, government of India. In this project work, I have worked to collect and expand what Ramanujan did with Nested Radicals and summarized all important facts into the one article. In the article, there are formulas, formulas and only formulas — I think this is exactly what Ramanujan is known for. This article not only deals with Ramanujan’s initial work on Nested Radicals but…

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…

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

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 \ \ldots (1)$ Replacing $ a$ by $ (n+a)$ where $ x, n, a \in \mathbb{R}$ , we can have $…

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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, said to be created by Pope Gregory XIII. Gregorian calendar was introduced in 80s of 16th century, to be accurate…