Proof problem about prime numbers

Question

1) The numbers 3, 5, 7 are all prime. Does it ever happen again that three numbers of the form n, n + 2, n + 4 are all prime? How to prove it?

2) There are four primes between 0 and 10 and between 10 and 20. Does it ever happen again that there are four primes between two consecutive multiples of 10? How to prove it?

Answer 1

All numbers not divisible by 3 can be represented as 3n±1, where n is an integer, simply because, of the two integers between each multiple of 3, one follows the multiple of 3 and the other precedes the next multiple.

If we add 2 we get either 3n+1+2=3n+3 (divisible by 3); or 3n-1+2=3n+1.

If 3n+1 is a prime number, then adding 2 to it will create a number divisible by 3.

If 3n-1 is a prime number, then adding 4 to it will create a number divisible by 3.

Therefore, with the exception of  3, 5, 7, you cannot have 3 successive odd numbers all of which are prime. And, of course, apart from 2, all even numbers are composite.