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).

 

[Architeuthis Dux]

The reason we use (mod3) in this solution is because it helps us determine the divisibility of the numbers by 3. When we say p=0 (mod3), it means that p leaves a remainder of 0 when divided by 3. Similarly, p=1 (mod3) means that p leaves a remainder of 1 when divided by 3, and p=2 (mod3) means that p leaves a remainder of 2 when divided by 3.

Using this method, we can quickly determine that at least one of the three consecutive odd numbers (p, p+2, p+4) is divisible by 3. And since we are looking for prime numbers, the only possible solution is when p=1 (mod3), which gives us the sequence 3,5,7.

Using (mod3) is just one way to approach this problem. You could also use other methods such as checking for divisibility by 2 or 5, but (mod3) is a common and efficient method to use in this case.