Vikram P

Prove that if (10^n-1)/9 = Prime then 'n' is also a prime number.

Recently through stumble upon I found "Wepware", a new google chrome extension. The cool part in my opinion about that extension is ability to capture any part of the website, and when you open that grab it shows realtime content of that relevant part only.

you can see this video to understand what I am referring to:

http://www.youtube.com/watch?v=pvTanWngkBg

Poll: How will you use this type of functionality?

Have a look at the following video

http://www.youtube.com/watch?v=vWVZ6APXM4w

What do you think is the reason?

I think angular momentum helps it to maintain the same course (same as the one hit in the middle) and hence displacement is constant. I may be completely wrong but that is what I think. ;)

In his famous work "Mysterium Cosmographicum" Kepler proposed a three dimensional model based on nested Platonic solids to explain the mainly two dimensional planetary orbital motions. Refer to the image below:

http://chapin.williams.edu/pasachoff/graphics/Kepl...

Question (of course this has nothing to do with Kepler's work) :

If the five platonic solids Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron respectively (in that order) are packed inside spheres of optimum radius (i.e. most efficient packing) and if the length of side of Tetrahedron is 1 unit (which is the innermost platonic solid), find out the radius of outermost Sphere.

To avoid any misunderstanding, the configuration goes like this:

Firstly a Tetrahedron of unit side length is packed inside a sphere, which is inside a Cube packed inside a sphere, which is again inside a Octahedron packed inside a sphere, and so on.. . . this ends which a Icosahedron being packed inside a sphere.

Checkout the following video

http://www.youtube.com/watch?v=1yaqUI4b974

Since we know frequency of each configuration, is there any way that we can find mathematical equation for a given configuration at a particular frequency?

For example what is the underlying function to the first configuration appearing at 15th second of the video at 368hz?

A prime number "P" is a Precious Prime of Order "N" if Σ Prime(i)^n (where Prime(i)=2 to P) is also a prime for all n = 1 to N

Let's take some example:

3 is a precious prime of order 2 because 2^2 + 3^2 = 13 which is a prime and 2^1+3^1=5 which is also a prime.

Like that 33773 is a precious prime of order 4 because

2^1 + 3^1 + 5^1 + 7^1 + 11^1 + . . . . . + 33773^1 = 57 584 381 which is a prime

2^2 + 3^2 + 5^2 + 7^2 + 11^2 + . . . . . + 33773^2 = 1 276 400 440 541 which is also a prime

2^3 + 3^3 + 5^3 + 7^3 + 11^3 + . . . . . + 33773^3 = 32 129 765 754 996 971 which is also a prime

2^4 + 3^4 + 5^4 + 7^4 + 11^4 + . . . . . + 33773^4 = 865 842 131 651 904 557 537 which is also a prime

Question:

Find next such Precious Prime of order 4. BTW I really doubt if we can have precious primes of order higher than 4 :)

Let PP denote sequence of perfect prime numbers which are derived from

[pq mod (p+q)] where p & q both are prime numbers.

For instance for p =2 and q=5 we have PP = (2*5) mod (2+5) = 3 which is a prime number so 3 is PP. Similarly for p=83 and q=71 we have PP = 5893 mod 154 = 41 which is again a prime number so 41 is also PP. There are many such examples.

Can you find NPP i.e. sequence of Non Perfect Prime numbers (numbers that can not be derived from prime numbers as discussed above) ?

Let S(n) (where n > 2) represented sum of the areas of 'n' sided regular polygon which are infinitely inscribed within each other where largest polygon has side length of 1 unit.

Refer to the following image

http://aiminghigh.aimssec.org/wp-content/uploads/2...

It shows inscribed squares and we know that S(4) = 2 which is convergent. I believe at certain large 'n', S(n) becomes divergent or at least inconclusive.

Question: At what 'n', S(n) becomes divergent? What are your thoughts on S(n)?

Let p represent a prime number and sd(p) denote sum of digits of 'p'.

Let S(n) represent sequence where [p^sd(p)+1] = 0 mod sd(p)

S(n) = 11, 37, 73, 101, 127, 163, 223, 263, 307, 337 . . .

Is there any way to define S(n) in any other terms or is there a way to find 'n'th term directly? Are there any interesting characteristics of this sequence?

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16.

What is the probability of a randomly selected number being

(1) an Abundant Number

(2) an Odd Abundant Number

Do you know that 23-March-2013 or 2332013 is a prime day. So let's celebrate it with the following question.

Let S denote sum of squares of first n primes. Prove that for n>1, S can not be a square.

(1) A sequence {R} of rectangles is generated as shown in the image below.

http://farm9.staticflickr.com/8088/8557307742_ed6e...

We begin with a unit square R₀ and adjoin the rectangle, of unit area on one SIDE to produce R₁. Then a a next step we adjoin the rectangle of unit area on TOP of the R₁ to produce R₂. If we continuously follow this process on side & top respectively of the previous figure, the ratio of length to height of the rectangle will converge to ______ ?

(2) ∏ (n=1 --> ∞) 4n²/(4n²-1) = _____ ?

Please answer these questions by tomorrow, otherwise there will be no fun ;-)

Have a look at following image:

http://farm9.staticflickr.com/8516/8537873172_b301...

The image shows top pose of a box with mirror arrangement (red line indicates reflectors)

(1) In the first arrangement will the light will ever come out (i.e. will it come out from the entry point or will it bounce back forever inside. In case someone is able to build a perfect infinite reflector what kind or image / scene will appear when someone from top looks at one of the reflectors?

(2) For an arrangement shown in the right hand side of the image,what results one will get if someone conducted double split experiment in that?

This is something I am very curious about, hope someone can help :)

Consider a number (N) generated using concatenating squares of consecutive primes.

N = (P₁)² (P₂)² (P₃)² . . . . . .(Pk)^2

Easy Question: For what values of "k" , "N" will always be a composite number? Why?

Tough Question: For k = 2, is there a simple way/logic to determine whether N is a prime number?

A number is called Sandwich Number if its neighbors are power numbers. i.e. "n" can be considered as Sandwich Number if n-1 and n+1 can be written in form of a^b.

From Fermat's Sandwich Theorem we know that 26 is the only number which comes in between square and a cube. i.e. 5^2 & 3^3.

Can you find any other?

Refer to the following question (and diagram) recently asked by falzoon

http://answers.yahoo.com/question/index?qid=201301...

Now imagine that in the picture you do not divide the square in three equal areas but you draw three parallel lines such that point A coincides with point G. Point B is on line CF and Point C is on line DE.

Questions:

(1) Is it possible to have such arrangement with a square? If yes, What is the distance between three segments?

(2) If this can be done only with rectangle then what is the dimension of such rectangle?

Show that, For positive integers a & b that are relatively prime If (a^n+b^n)/(a+b) is a prime then "n" is also a prime.