Gaps between prime numbers represent an interesting and widely
studied topic. Taking the first few primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
there are gaps between them
1, 2, 2, 4, 2, 4, 6, ...
We see that the smallest gap 1 is an exception: regardless
of this, gaps are always positive even numbers. The gap 2 is between so-called
prime twins, the smallest such twins are 3 and 5. The prime twins may be
infinitely many, but proof of this has not yet been given.
The largest gap does not exist because a gap can be of any
length. This follows from this finding: if we take any natural number n,
the numbers n! +2, n! +3, ..., n! + n are all
composed (surely, n! +2 is divisible by two, n! +3 is divisible
by three, and so on) and is therefore a part of some gap between prime
numbers.
As already mentioned, primes 3, 5 are smallest consecutive
primes with gap 2. Furthermore:
· Prime numbers 5, 7, 11 are the smallest consecutive primes
with gaps 2, 4.
· Prime numbers 17, 19, 23, 29 are the smallest consecutive
prime numbers with gaps 2, 4, 6.
· Prime numbers 347, 349, 353, 359, 367 are the smallest
consecutive prime numbers with gaps 2, 4, 6, 8.
· Prime numbers 13901, 13903, 13907, 13913, 13921, 13931 are
the smallest consecutive prime numbers with gaps 2, 4, 6, 8, 10.
In this way, we could continue up to a string of fifteen
consecutive prime numbers, each with by two increasing gaps from 2 to 28, which
is the last result of this kind. The first prime
number here is 221860944705726407. Finding the following string of 16 consecutive primes with gaps from 2 to 30 can
be a challenge for the reader (Puzzle 2). See OEIS
A016045.
Let us look at four consecutive prime numbers 17, 19, 23,
29, which are the smallest with gaps 2, 4 and 6. If we calculate the sum of
squares
172
+ 192 + 232 + 292,
the result is 2020. Pour
féliciter 2020.
