Multiple scenarios

Number Theory Level 2

Let $P>3$ be a prime number. Which of the following is always false?

p + 2

is prime

p + 4

is prime neither is prime both

p + 2

and

p + 4

are prime

2 solutions

Jonathan Wong
Nov 15, 2014

$P$ , $P+2$ and $P+4$ are distinct modulo 3, so by the pigeonhole theorem, exactly one must be divisible by 3, and therefore composite (for $P>3$ ). Thus not all of $P$ , $P+2$ and $P+4$ can be prime.

what is pigeon hole theorem

Guru Raj - 6 years, 6 months ago

https://brilliant.org/discussions/thread/pigeonhole-principle-2/

Jonathan Wong - 6 years, 6 months ago

Your explanation is exact.

Ram Hegde - 6 years, 6 months ago

Seth Lovelace
Nov 13, 2014

Assume we have a prime number greater than three. Lets call this number P. This number must logically be an odd number. If we look at the odd numbers, one can see that:

5, 7, 9, 11, 13, 15, 17, 19, 21, ...

A multiple of three arises every 3 terms after 5, thus either P+2 and P+4 must be a multiple of three. The only exception to this rule is 3, 5, and 7, but since P > 3, this is now impossible. Thus there exists no chain of primes P, P+2, and P+4 that are all prime.

If I say , p=11 , then how is it possible?

Selim Hossain - 6 years, 6 months ago

agreed take 13 =p ie p+2 will not satisfy take p=17 p+4 will not satisfy...

Gunanjan Singh - 6 years, 6 months ago

Good evidence.

Ram Hegde - 6 years, 6 months ago

let p=5;then p+2 =7& p+4=9 which is not prime beacuse 9/3=3

Hamza Hassan - 6 years, 6 months ago

When I said the only exception is 3, 5, and 7, I was referring to that sequence of numbers, i.e. when p=3.

Seth Lovelace - 6 years, 6 months ago

Multiple scenarios

2 solutions

0 pending reports