three primes?

Question

prove that it is impossible to have three consecutive intergers all of which are prime.

Answer 1

For any 3 consecutive numbers, one of them is divisible by 3. The only prime divisible by 3 is 3, so the only possibilities are 1, 2, 3; 2, 3, 4; 3, 4, 5. Since 1 is not a prime and 4 is not a prime none of these sequences are all prime, so there are no such sequences of primes.

Answer 2

If you define 1 as prime then you can have 3 consecutive integers that are prime otherwise

let n be prime.
n, n+1, n+2.
if n is even then n=2 and n+2 is even which makes it divisible by 2 which is not prime.
if n is odd then n+1 is even and again is not prime.

Answer 3

I won't give you a mathamatical proof because it takes time to write out, and I don't know if you are at a math level which would require the type of proof I learned in university.

But the reason is in every set of three consecutive numbers there is at least one even number. ex: 3, 4, 5 or 4, 5, 6. In the first example there is one even number and in the second example there are two even numbers. This will always be the case. There will always be 1 or 2 even numbers.

By definition even numbers are not prime. An even number is always devisable by 2. Which means it is not a prime, since by defintion primes are only divisable by the number 1 and themselves.

Answer 4

A prime is only divisible by itself and 1. Think of three consecutive integers. Is one of them even? Then it can't be prime, except for 2, but 1 is not prime.

Answer 5

1 3 11

Answer 6

2 is a prime number, every other number is divisible by 2 [all even numbers], therefore every other number except 2 cannot be prime because every even number is divisible by 2.

With exception to the numbers 2 and 3, you will find that it is impossible to have even 2 consecutive integers next to each other that are prime.

Answer 7

Since the term prime is meaningless unless the integer is greater than one, consider any sequence of three integers greater than one. At least one of these will be divisible by 2 and will not be 2. Thus, at least one is not prime, and therefore no sequence of three integers greater than one can all be prime.

If we need algebra, first assume the smallest integer is even. Then we have integers 2n, 2n+1, and 2n+2 (n>=1 and an integer). 2n+2 = 2(n+1) is not prime.

If instead the first integer is odd, we have 2n+1, 2n+2, and 2n+3, n>=1. Again, 2n+2 = 2(n+1) is not prime.

Answer 8

Lessee: a prime number can't be divided evenly by any number except itself or 1. How about 1, 2, and 3?

Answer 9

At least one of those numbers would have to be even and all even numbers are divisible by two (half) and therefore not prime.

Answer 10

you cant cause one will even up being even, thus is divisible by 2. thus not prime.