Puzzle 2: Happy New Year 2020 (and the link with 2- 4- 6- concecutive prime gaps)

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.