Sequences and series

Question

anonymous · Answer

Prove that the number 111....111(91digits) is not a prime number

ganeshie8 · Answer

1 = 10^0
11 = 10^0 + 10^1
111 = 10^0 + 10^1 + 10^2
... 
11111...(91 digits) = 10^0 + 10^1 + 10^2 + ... + 10^90

anonymous · Answer

So how does it prove

ganeshie8 · Answer

find the sum using partial sum formula for geometric series

anonymous · Answer

How does it prove it is not a prime number

ganeshie8 · Answer

not yet, lets find the sum and it may give us some clue

anonymous · Answer

sum of

ganeshie8 · Answer

11111...(91 digits) = 10^0 + 10^1 + 10^2 + ... + 10^90


notice that this is a geometric series
first term = 10^0
common ratio = 10

ganeshie8 · Answer

total number of terms = 91

anonymous · Answer

yeah got how do then

ganeshie8 · Answer

use the partial sum formula :

$$\large   11111...(91 digits) = \dfrac{10^{91}-1}{10-1}$$

ganeshie8 · Answer

$$\large  =  \dfrac{10^{13\times  7}-1}{9}$$

anonymous · Answer

then it proves it is not a prime number nice !

ganeshie8 · Answer

how ?

anonymous · Answer

No it doesn't i was thinking about taking 1/9 common

vishweshshrimali5 · Answer

It does :)

anonymous · Answer

I wil have to go for college , bye for now

ganeshie8 · Answer

recall below formula :
$$x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y+ \cdots  + y^{n-1})$$

vishweshshrimali5 · Answer

Sorry reverse the values:

$$\large{x = 10^7; y= 10^7; n =13}$$

vishweshshrimali5 · Answer

I would never know why I making these mistakes:

$\color{blue}{\text{Originally Posted by}}$ @vishweshshrimali5
Sorry reverse the values:

$$\large{x = 10^7; y= \color{red}{1}; n =13}$$
$\color{blue}{\text{End of Quote}}$

anonymous · Answer

@vishweshshrimali5 are you alright??

ganeshie8 · Answer

$$\large \dfrac{10^{13\times  7}-1}{9} =  \dfrac{(10^{13})^7-1^7}{9}$$

$$\large    = \dfrac{1}{9}(10^{13}-1)[(10^{13})^6 + (10^{13})^5 + \cdots + 1]$$

vishweshshrimali5 · Answer

This is the main problem - I am ill today :(

anonymous · Answer

Here, divisibility by 9 does not prove that number is not prime??? :P

ganeshie8 · Answer

exactly !

9 divides $10^{13}-1$, leaving us a product of two numbers none of which are 1

anonymous · Answer

@ganeshie8  from then, can we continue break $10^{13}-1=(10-1) *(somethingelse)$ to have the first term =9 to cancel out with 1/9. That makes the 111...... becomes a product of 2 numbers--> it's not a prime. Am I right?

ganeshie8 · Answer

yes that looks perfect to me !

anonymous · Answer

Wow, you are genius on number theory field. hihihi

ganeshie8 · Answer

lol no ways, but ty :)

kainui · Answer

Wow awesome. Another reason to love geometric series.