Final answer:
The question seeks to find a prime triplet, but after the set (3, 5, 7), there is no such triplet where n, n+2, and n+4 are all primes, because one of the numbers will be divisible by 3, thus it cannot be prime.
Step-by-step explanation:
The question asks to show that there exists an integer n such that n, n + 2 and n + 4 are all primes. However, the information provided does not directly relate to proving the existence of such a prime triplet. In general, a prime triplet is a set of three prime numbers in which the smallest and largest numbers differ by four.
The smallest and best-known example of a prime triplet is (3, 5, 7). After this triplet, the number of the form n, n + 2, and n + 4 cannot all be prime because one of them will be a multiple of 3. If n is not divisible by 3, then either n + 3 is, making n + 4 not prime, or n + 2 or n itself is divisible by 3.
To show that there exists an integer n such that n, n + 2, and n + 4 are all prime numbers, we can use the concept of prime numbers and the properties of addition.
Let's consider the number n. If n is an even number, then n + 2 will also be even, making it not prime. Therefore, n must be odd.
We can start by assuming n = 3, which is an odd number. Checking the values of n, n + 2, and n + 4:
n = 3, n + 2 = 5, n + 4 = 7.
All three numbers are prime numbers, proving that there exists an integer n (in this case, n = 3) where n, n + 2, and n + 4 are all prime numbers.