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# Monthly Archives: September 2011

## Into the Mandelbrot set

The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognizable two-dimensional fractal shape. More detail on Wikipedia..

Mandelbrot Fractal

## LOVE Exists

The plot of the equation $\sqrt{\left({1-(|x|-1)^2}\right)}=-2.5 \sqrt{\left( {1-\sqrt{\dfrac{|x|}{2}}}\right)}$ and following other equations are heart curves:

# Proof

Images via Wolframalpha.com and i.imgur.com respectively.

## Fermat Numbers

Fermat Number, a class of numbers, is an integer of the form $F_n=2^{2^n} +1 \ \ n \ge 0$.

For example: Putting $n := 0,1,2 \ldots$ in $F_n=2^{2^n}$ we get $F_0=3$, $F_1=5$, $F_2=17$, $F_3=257$ etc.

Fermat observed that all the integers $F_0, F_1, F_2, F_3, \ldots$ were prime numbers and announced that $F_n$ is a prime for each natural value of $n$.

In writing to Prof. Mersenne, Fermat confidently announced:

I have found that numbers of the form $2^{2^n}+1$ are always prime numbers and have long since signified to analysts the truth of this theorem.

However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that $F_1 -F_4$ are primes but $F_5=2^{2^5} =4294967297$ is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that $F_5$ is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.

# Theorem:

The Fermat number $F_5$ is divisible by $641$ i.e., $641|F_5$.

# Proof:

As defined $F_5 :=2^{2^5}+1=2^{32}+1 \ \ldots (1)$

Factorising $641$ in such a way that $641=640+1 =5 \times 128+1 \\ =5 \times 2^7 +1$
Assuming $a=5 \bigwedge b=2^7$ we have $ab+1=641$.

Subtracting $a^4=5^4=625$ from 641, we get $ab+1-a^4=641-625=16=2^4 \ \ldots (2)$.

Now again, equation (1) could be written as
$F_5=2^{32}+1 \\ \ =2^4 \times {(2^7)}^4+1 \\ \ =2^4 b^4 +1 \\ \ =(1+ab-a^4)b^4 +1 \\ \ =(1+ab)[a^4+(1-ab)(1+a^2b^2)] \\ \ =641 \times \mathrm{an \, Integer}$
Which gives that $641|F_n$.

Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond $F_4$. The best guess is that all Fermat numbers $F_n>F_4$ are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers $F_n, F_m \ m > n \ge 0$, $\mathrm{gcd}(F_m, F_n) =1$.

Following two theorems are very useful in determining the primality of Fermat numbers:

# Pepin Test:

For $n \ge 1$, the Fermat number $F_n$ is prime $\iff 3^{(F_n-1)/2} \equiv -1 \pmod {F_n}$

# Euler- Lucas Theorem

Any prime divisor $p$ of $F_n$, where $n \ge 2$, is of form $p=k \cdot 2^{n+2}+1$.

Fermat numbers ($F_n$) with $n=0, 1, 2, 3, 4$ are prime; with $n=5,6,7,8,9,10,11$ have completely been factored; with $n=12, 13, 15, 16, 18, 19, 25, 27, 30$ have two or more prime factors known; with $n=17, 21, 23, 26, 28, 29, 31, 32$ have only one prime factor known; with $n=14,20,22,24$ have no factors known but proved composites. $F_{33}$ has not yet been proved either prime or composite.

## Three Children, Two Friends and One Mathematical Puzzle

Two close friends, Robert and Thomas, met again after a gap of several years.
Robert Said: I am now married and have three children.
Thomas Said: That’s great! How old they are?
Robert: Thomas! Guess it yourself with some clues provided by me. The product of the ages of my children is 36.
Thomas: Hmm… Not so helpful clue. Can you please give one more?
Robert: Yeah! Can you see the number on the house across the street?
Thomas: Yes! I can.
Robert: The sum of their ages equal that number.
Thomas: Sorry! I still could not determine their ages.
Robert: My oldest child has red hair.
Thomas: OH.. Oldest one? Finally I got it. I know age of each of your children.

# Question:

What were the ages of Robert’s children and how did Thomas know?

This is a very good logical problem. To do it, first write down all the real possibilities that the number on that building might have been. Assuming integer ages one get get the following which equal 36 when multiplied:

 Age of 1st Age of 2nd Age of 3rd Sum(HouseNo.) 1 1 36 38 1 2 18 21 1 3 12 16 1 4 9 14 1 6 6 13 2 2 9 13 2 3 6 11 3 3 4 10

The biggest clue is that the Thomas DID NOT KNOW after having been told the sum equaled the number on the house. Why didn’t he know? The only reason would be that the number was 13, in which case there are two possible answers. For any other number, the answer is unique and the Thomas would have known after the second clue. So he asked for a third clue. The clue that the oldest had red hair is really just saying that there is an “oldest”, meaning that the older two are not twins. Hence, the answer is that the redhead is 9 years old, and the younger two are both 2 years old.

Source of The Puzzle: This puzzle is a modified form of a puzzle from Science Reporter Magazine, Hindi 1996 and I have changed the names from Ram and Shyam to Robert and Thomas to make this puzzle convenient to read.

## How to Draw the Famous Batman Curve

The ellipse $\displaystyle \left( \frac{x}{7} \right)^{2} + \left( \frac{y}{3} \right)^{2} - 1 = 0$ looks like this:

So the curve $\left( \frac{x}{7} \right)^{2}\sqrt{\frac{\left| \left| x \right|-3 \right|}{\left| x \right|-3}} + \left( \frac{y}{3} \right)^{2}\sqrt{\frac{\left| y+3\frac{\sqrt{33}}{7} \right|}{y+3\frac{\sqrt{33}}{7}}} - 1 = 0$ is the above ellipse, in the region where $|x|>3$ and $y > -3\sqrt{33}/7$:

That’s the first factor.
(more…)

## Pi (Π)

Pi, the mathematical constant, has a yet to determine value. As of now following sequence of digits has found to be most accurate for the approximation for $\pi$. :3.1415926535897932384626433832795028841971693993751058209 7494459230781640628620899862803482534211706798214808651 3282306647093844609550582231725359408128481117450284102 7019385211055596446229489549303819644288109756659334461 2847564823378678316527120190914564856692346034861045432 6648213393607260249141273724587006606315588174881520920 9628292540917153643678925903600113305305488204665213841 4695194151160943305727036575959195309218611738193261179 3105118548074462379962749567351885752724891227938183011 9491298336733624406566430860213949463952247371907021798 6094370277053921717629317675238467481846766940513200056 8127145263560827785771342757789609173637178721468440901 2249534301465495853710507922796892589235420199561121290 2196086403441815981362977477130996051870721134999999837 2978049951059731732816096318595024459455346908302642522 3082533446850352619311881710100031378387528865875332083 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