How to find one prime factor of $\large 2^{23}-1$ without using computer

Question

How to find one prime factor of $\large  2^{23}-1$  without using computer

anonymous · Answer

you can use https://www.mathway.com/

acxbox22 · Answer

"without using computer"

anonymous · Answer

oh oops

acxbox22 · Answer

@ganeshie8 d you know of a way?

ganeshie8 · Answer

I'm still reading the material, but I know there is a nice way to work for numbers of this form : $$\large 2^p-1$$ Euler proved $\large 2^{31}-1$ is prime back in the days when electricity itself was not known https://primes.utm.edu/notes/by_year.html

ikram002p · Answer

never seen this before  @rational

rational · Answer

here are some results i know so far :

1) $\large 2^a-1$ is a prime  $\large \implies  $  $\large  a$ is a prime
2)  $\large a$ is composite $\large  2^a-1$ is composite
3)  odd prime factors of $\large  2^p-1$ will be of form $\large  2kp+1$

rational · Answer

the first two results are same  :- contrapositives of each other

rational · Answer

the last result is very useful in finding prime factors of 2^p-1

rational · Answer

for example, finding prime factors of $\large 2^{23}-1$ is extremely trivial because we know that they will be of form $\large   2k*23 - 1 = 46k+1 $

ikram002p · Answer

interesting O.O

rational · Answer

typo :

$\large  2k*23\color{Red}{+}1 =  46k+1$

ikram002p · Answer

i should read more of these stuff

rational · Answer

but ofcourse not all numbers of this form are factors of 2^23-1

you need to check if the numbers really divide 2^23-1 or not

rational · Answer

you need to check if   47 | 2^23-1 or not

ikram002p · Answer

yess yess but it well reduce possible factors to
(1/2)*(1/46) for all n <2^23-1

ikram002p · Answer

will*

rational · Answer

yes

ikram002p · Answer

cool ^^

rational · Answer

it reduces by 2p only

rational · Answer

also very few numbers of form 2kp+1 will be prime
so that eliminates lot of trials

ikram002p · Answer

wait  i took 1/2 from sqrt n  thingy :o
which will already reduce them to half
then as u said 2p
also factorz will be >2p

rational · Answer

yeah reducing by sqrt(n)  is not same as reducing by 2

rational · Answer

sqrt(n) <<<<<<<   n/2

rational · Answer

sqrt(36) <<<<< 36/2

ikram002p · Answer

ohk :o

rational · Answer

its an interesting proof to see why odd factors of 2^p-1 will be of form 2kp+1
i have posted this question in MSE the other day

rational · Answer

go through when free

rational · Answer

*odd prime factors of 2^p-1 will be of form 2kp+1

ikram002p · Answer

its really interesting as i said i never seen it before