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2017 is not just another prime number

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Re: 2017 is not just another prime number

#31

Post by indexerror »

kondbg wrote: post #30

> 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?

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

Re: 2017 is not just another prime number

#32

Post by cperciva »

ComodoHacker wrote: post #29
williamstein wrote: post #17

Verifications of all the statements using SageMath, in case you want to be convinced or explore further: https://cloud.sagemath.com/projects/4a5f0542-5873-4eed-a85c-...

> 2017 can be written as a sum of cubes of five distinct integers. This gives no results in SageMath...

The greedy algorithm works here: 2017 = 12^3 + 6^3 + 4^3 + 2^3 + 1^3

Re: 2017 is not just another prime number

#34

Post by gus_massa »

cperciva wrote: post #32
ComodoHacker wrote: post #29

Earlier quoted context omitted.

> 2017 can be written as a sum of cubes of five distinct integers. This gives no results in SageMath...

The greedy algorithm works here: 2017 = 12^3 + 6^3 + 4^3 + 2^3 + 1^3

Technical note: 5 cubes is not enough for every number, so this property of 2017 is not trivial.

From

> Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

> 23 = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3.

I couldn't find if it's common that 5 cubes is enough. [This looks like a nice exercise for the reader.]

Re: 2017 is not just another prime number

#35

Post by tjwei »

ComodoHacker wrote: post #29
williamstein wrote: post #17

Verifications of all the statements using SageMath, in case you want to be convinced or explore further: https://cloud.sagemath.com/projects/4a5f0542-5873-4eed-a85c-...

> 2017 can be written as a sum of cubes of five distinct integers. This gives no results in SageMath...

And 2017 = 2^3 + 4^3 + 6^3 + 9^3 + 10^3

Re: 2017 is not just another prime number

#36

Post by aisofteng »

indexerror wrote: post #31
kondbg wrote: post #30

> 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?

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

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

Re: 2017 is not just another prime number

#37

Post by dwe3000 »

mikeash wrote: post #8
gcr wrote: post #6

Earlier quoted context omitted.

Every number is special. Proof is by contradiction: Assume that not every number is mathematically interesting and let X be the first such number. However, the fact that X is the lowest such number is itself pretty special, right?

Can this proof be adapted for the reals, or is it only the case that every integer is special?

Outside the set of integers, the only reals left are either only fractionally interesting or crazy

Re: 2017 is not just another prime number

#38

Post by aisofteng »

joeax wrote: post #23

Meh. A prime number year last happened in 2011. Just kidding... happy new prime number year! BTW I'm really looking forward to the next perfect square year: 2025 (45^2). It last happened in 1936, and won't happen again until 2116.

It would be an interesting exercise to consider the years that are a perfect square n^2 and then try to give some sort of attempt at dividing history into n segments of n years, trying to find a common theme for each segment.

Re: 2017 is not just another prime number

#39

Post by aisofteng »

gcr wrote: post #6
Houshalter wrote: post #5

Earlier quoted context omitted.

What I like to do to measure the "mathematical interestingness" of a number, is check how many times it appears in OEIS. A database of sequences of numbers found in mathematical research. 2017 appeared in 453 sequences. For comparison; 2016 appears 833 sequences, and 2018 appears in 113. http://oeis.org/search?q=seq%3A2016&sort=&language=english&g...

Every number is special. Proof is by contradiction: Assume that not every number is mathematically interesting and let X be the first such number. However, the fact that X is the lowest such number is itself pretty special, right?

This "proof" seems to be popular, and it always bothers me that it's invalid. It uses self-reference in an invalid way.

Allow me to formalize; we take as a rigorous definition of an "interesting number" that a number has a unique property. Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. In formal first order logic, n is interesting if there exist a predicate P and a number n such that P(n) is true and if m != n then P(n) is false.

Let I, as a subset of the natural numbers N, be the set of interesting numbers. There are two cases: either N - I is empty, or it is not. If it is not, let n be the least element of N - I. n is therefore interesting, having a unique property in that it is the smallest integer not in I; however, this is a contradiction, because we defined I to include all interesting numbers, and so N - I must be empty; in other words, every number is interesting.

Edit: Actually, my definition of "interesting" seems to be in second order logic [1], since I'm using an existential quantifier for predicates. It doesn't seem possible to give a definition of this sense of "interesting" in first order logic.

[1] https://plato.stanford.edu/entries/logic-higher-order/

Re: 2017 is not just another prime number

#40

Post by Someone »

aisofteng wrote: post #36
indexerror wrote: post #31

Earlier quoted context omitted.

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

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…

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