The numbers in the list are all prime but they don't all differ by 2. 45, 49 and 51 are missing and are composite numbers. As numbers get larger the gaps between primes can be quite large.
To prove whether a number is prime you take the rough square root.
41, 43, 47 all have square roots between 6 and 7; 53 and 61 have square roots between 7 and 8.
It's only necessary to test divisibility by the prime numbers up to the approximate square roots.
So for 41, 43, 47 the prime number divisors are 2, 3, 5; and for 53 and 61 we also need to include 7. Since none of these primes are factors of the listed primes, the listed numbers are in fact prime.
Prime numbers can be created using certain formulas, but there is no simple test to determine if a particular number is prime, although there are algorithms (usually run in computers) that can carry out a whole range of tests.