Question

\[\sigma(n+1)-\sigma(n-1)=2\]

Answer 1

twin primes?

Answer 2

what is a question? :)

Answer 3

idk if this is true for anything other than twin primes

Answer 4

Wow, Is that too easy ha !

Answer 5

can you specify more clearly sigma and n?

Answer 6

\(\sigma(n) = n+1 \iff \text{ n is prime}\)

Answer 7

yes, I see that but I'm trying hard to check why it cannot be something of this form\[\sigma(n+1) = n+1 + k\]\[\sigma(n-1) = n-1 + k\]that is to say that the proper divisors have the same sum. let me think.

Answer 8

Interesting...

Answer 9

:/ what we need to do in this question tho?..

Answer 10

@sparrow2 \(\sigma\) is the sum of divisors function
http://mathschallenge.net/library/number/sum_of_divisors

Answer 11

sigma function is defined for positive integers and its minimum value is 1
and for \(\sigma(x)=1\) ; x=1

so for \(\sigma(n-1)+\sigma(n+1)=2\)
both (n-1) and (n+1) must be 1 and this is not possible

Answer 12

you answered a different question lol

Answer 13

which question are we solving?

Answer 14

so we are trying to prove that sigma(n)=n+1 is working only for primes?

Answer 15

ok I will write a quick program to find non-primes with the same property

Answer 16

@ParthKohli 435 is the first counter example it seems

Answer 17

oh you already did it. great :)

Answer 18

http://www.wolframalpha.com/input/?i=sigma(435%2B1)+-+sigma(435-1)

Answer 19

Are there infinitely many counter examples?

Are there nontwin primes that satisfy the equation AND also satisfy this equation?

\[\tau(n+1)=\tau(n-1)\]

Answer 20

if you plug n=4 : sigma(5)-sigma(3)=6-4=2.. is 4 also solution?

Answer 21

Yeah, it definitely is @sparrow2 :D

Answer 22

then every n,there n+1 and n-1 are primes are solutions too,there are many solution

Answer 23

Yeah, when \(n+1\) and \(n-1\) are primes, these are called twin primes, so you just found 3 and 5 are twin primes, another pair of twin primes is 11 and 13.

Answer 24

less then 1000 there are 36 solutions :D

Answer 25

if my program works well :D i'm sure it works

Answer 26

are twin primes infinite?

Answer 27

oh ist's unsolved :D

Answer 28

Ok so well you might have found it yourself because that formula works well for prime numbers only....