The only 3 consecutive odd numbers that are primes are 3,5,7

[CmbkG]

Homework Statement

Show that the only three consecutive numbers that are primes are 3,5,7.

Homework Equations

The Attempt at a Solution

let p, p+2, p+4 be three consecutive odd numbers
If p=0(mod3), p is divisible by 3
If p=1(mod 3), p+2 is divisible by 3
If p=2(mod3), p+4 is divisible by 3

This means at least one of p, p+2, p+4 is divisible by 3

Since we are looking for prime numbers 3 can be the only number that is divisible by 3. Therefore we only have 3 possible solutions:

-1,1,3
1,3,5
3,5,7

Since -1 and 1 are not primes the only possible solution is 3,5,7


-I no i have the solution here, its just i was helped with this and i don't quite understand why we bring in (mod3) is that just the way it is done or why do you include it??

 

[willem2]

One way of proving that a number (or at least one of 3 numbers) isn't prime, is proving that it is divisible by another prime. We know the numbers are odd, so 3 is the next candidate.

One way of proving a result concerning divisibility by a particular number, is to consider all cases modulo that number, in this case p=0,1,2 (mod 3).