All whole numbers, integers, are divisable by 1 and themselves. Prime numbers are ONLY divisible by 1 and themselves.
Example,
None prime,
8 (1x8, 2x2x2, 2x4, 8x1)
539 (1x539, 11x49, 11x7x7, 539x1)
Prime
3 (1x3, 3x1)
181 (1x181, 181x1)
Note
1 is not considered to be a prime, so the smallest prime number is 2, which is also the only prime number that is even, as all even numbers greater than 2 are also divisible by 2.
Prime numbers less than or equal to 200.
Primes that differ by two, (3,5) are twins, and three numbers that differ by 2 (3,5,7) are triplets.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151,
157, 163, 167, 173, 179, 181, 191, 193, 197, 199
Assume the contrary, there is a finite number of primes (1)
(2, 3, 5, 7.........a, b, c, d.......n)
Let P be the product all these primes such that
P = (2)(3)(5)(7)......(a)(b)(c)(d)........(n) (2)
Add 1 to both sides
P + 1 = (2)(3)(5)(7)......(a)(b)(c)(d)........(n) +1 (3)
P + 1 cannot be a prime because of (1) and so must have at least one factor from the list of all primes shown in (2). It can be any prime but we will choose c.
Recall that for y = x + z, if y is divisible by p both x and z must be divisible by p. Here we are only considering integers so dividing must leave no remainders.Therefore in (3) both
(2)(3)(5)(7)......(a)(b)(c)(d)........(n) and 1 must be divisible by c,
but 1 is not divisible by any integer other than 1.
So statement 1 cannot be true and so there must be an infinite number of primes.
Note.
As there is an infinite number of primes it is impossible to perform step (2).
Prime numbers that differ by 2 are called prime twins. There are countless examples, (3 5) (11 13) (197 199). Euclid way back thought there was an infinite number but to date no one has been able to prove this. This is known as The Twin Prime Conjecture. Some progress has been made in recent decades but a final proof still seems a long way off. See this Britannica site for a brief summary.
There is an answer to this and it is an emphatic no. The following is based upon the excellent book "The Mathematical Universe" by William Dunham.
A prime triplet will be of the form p, p+2, p+4, such as 3, 5, 7.
Now any number divided 3 will have remainders of 0, 1 or 2.
For remainder of 0, then p/3 is a whole number, so p can be written 3k, where k is an integer. If k = 1, then p=3 and we have the triplet 3, 5, 7. But if k >= 1 then p = 3k is not a prime.
For remainder of 1, then p can be written 3k+1, where k is an integer. If k=0, then p=1 which is not a prime. So k >= 2. The second prime in the triplet is p+2 = 3k + 1 +2 = 3k + 3 = 3(k +1) and so not a prime. So no prime triplets when remainder is 1.
For remainder of 2, then p can be written 3k+2, where k is an integer k >= 0. The third prime in the triplet is p+4 = 3k + 2 +4 = 3k + 6 = 3(k +2) and so not a prime. So no prime triplets when remainder is 2.
So 3, 5, 7 is the only triplet!
Observation of primes show that to date there is no obvious exact pattern to their distribution, sometimes there are clusters where they are close together, whereas elsewhere there are large stretches where none appear. But you can easily find long sequences where there are no primes with the help of factorials.
Consider 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800
10! + 2 will be divisible 2 and so is not a prime
10!+ 3 will be divisible 3 and so is not a prime
. ....
10! + 9 will be divisible 9 and so is not a prime
10! + 10 will be divisible 10 and so is not a prime
So
3,628,802 3,628,803 3,628,804 3,628,805 3,628,806 3,628,807 3,628,808 3,628,809 3,628,810
is a sequence of 9 numbers which are not primes.
There may well be other sequences of 9 numbers which are not primes before this, just by inspecting the prime numbers less than 200 (see above) we find 114 to 126 is a sequence of 13, 140 to 148 and 182 to 190 are both exactly 9.
The same approach could be used for 50! to identify a sequence of 49 numbers that are not prime, and to identify these by inspection of numbers alone would be a large task. However be aware that 50! is the very large number
30414093201713378043612608166064768844377641568960512000000000000
For 500! to identify a sequence of 499 numbers that are not prime, but 500! is having 1135 digits!
But the general rule is a sequence of n none prime numbers can be generated by
(n+1)! + 2 (n+1)! +3 (n+1)! + 4 ..... (n+1)! + (n-1) (n+1)! + n
Thankfully it is impossible to compute infinity factorial .