[kondbg]

> The sum of the cube of gap of primes up to 2017 is a prime number. That is (3-2)^3 + (5-3)^3 + (7-5)^3 + (11-7)^3 + ... + (2017-2011)^3 is a prime number.

For the non-mathematically inclined, how do mathematicians come up with these? Are these just observations that they happened to witness, or are there underlying theoretical properties that allow one to derive this claim?

[indexerror]

There has been a great amount of research to find ways to check if a number of prime or not in polynomial time[1]. Many of such _facts_ are a observations from this conquest. Number theory reveals fascinating facts about spacing in prime numbers determining properties within a range. Sometimes such results emerge from there.

1: https://en.wikipedia.org/wiki/Primality_test

[aisofteng]

To expand in a way that answers the grandparents' question, part of the mentioned "great amount of research" came about from people looking for patterns anywhere they can, and patterns can be anywhere.

Someone thought to check how often there is a number and that number plus 2 that are both prime, and there seems to be a pattern there, which is the twin primes conjecture [1]. Along the way, a lot of other places are investigated in this search for patterns, such as the sum of the cube of gap primes that the grandparent mentions.

Recording investigations made along these lines is often done by recording it in the Online Encyclopedia of Integer Sequences [2]. (Significant findings merit publication in journals.)

The end result is that one can perform a search for a particular number and see in which sequences it appears. This is how the linked post came to be.

[1] https://en.wikipedia.org/wiki/Twin_prime#Conjectures [2] oeis.org

[Someone]

Also, searching oeis is easy: http://oeis.org?q=2017 currently produces 528 sequences containing the number 2017.