
2017 is not just another prime number - tjwei
http://weijr-note.blogspot.com/2017/01/2017-is-not-just-another-prime-number.html
======
williamstein
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-...](https://cloud.sagemath.com/projects/4a5f0542-5873-4eed-a85c-a18c706e8bcd/files/support/2017.sagews)

~~~
ComodoHacker
> 2017 can be written as a sum of cubes of five distinct integers.

This gives no results in SageMath...

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

~~~
gus_massa
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.]

~~~
Someone
[http://oeis.org/A003328](http://oeis.org/A003328): Numbers that are the sum
of 5 positive cubes

5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 64, 68, 71, 75, 78, 82, 83, 89,
90, 94, 96, 97, 101, 108, 109, 115, 116, 120, 127, 129, 131, 134, 135, 136,
138, 143, 145, 146, 150, 152, 153, 155, 157, 162, 164, 169, 171, 172, 176,
181, 183, 188, 190, 192, 194

It seems this is fairly common (1757 is the 1000th such number), but of course
that says nothing.

Reading
[http://mathworld.wolfram.com/CubicNumber.html](http://mathworld.wolfram.com/CubicNumber.html),
it is true that every sufficiently large integer is a sum of no more than 7
positive cubes.

It also states _”the only integers requiring nine positive cubes are 23 and
239. Wieferich proved that only 15 integers require eight cubes: 15, 22, 50,
114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, and 454 (OEIS
A018889).”_

Even stronger (same page): _”Deshouillers et al. (2000) conjectured that
7373170279850 is the largest integer that cannot be expressed as the sum of
four nonnegative cubes”_ (nice title for a paper: _”7 373 170 279 850.”_. See
[http://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-0...](http://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-01116-3/S0025-5718-99-01116-3.pdf))

If that is true, it is indeed common that 5 cubes is enough (since 4 almost
always would be sufficient)

~~~
ahakki
yes, but 2017 is the sum of five _destinct_ integer cubes

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joeax
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.

~~~
ClashTheBunny
I'm hoping to to see the only power of two in a millennium: 2048

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

~~~
Someone
Also, searching oeis is easy: [http://oeis.org?q=2017](http://oeis.org?q=2017)
currently produces 528 sequences containing the number 2017.

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JshWright
Here are 17 facts about 2017 in 2:17 from Matt Parker:

[https://www.youtube.com/watch?v=z6jMU-
AwX34](https://www.youtube.com/watch?v=z6jMU-AwX34)

(Some repeats, but plenty of non-prime facts as well (plus Matt's excellent
dry humor))

~~~
taneq
Huh wow, I went to uni with that guy!

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ChuckMcM
They forgot to calculate how many certificates on the Internet use 2017 as one
of their primes :-)

~~~
vog
Or, how many use a prime <= 2017, for that matter.

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kahrkunne
What I always wonder though, is 2017 really a special number or can you fit
things like this to every number?

~~~
Houshalter
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...](http://oeis.org/search?q=seq%3A2016&sort=&language=english&go=Search)

~~~
gcr
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?

~~~
aisofteng
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/](https://plato.stanford.edu/entries/logic-higher-order/)

~~~
zodiac
> Specifically, a number n is interesting if there is some predicate P(x)
> which is true only for n.

Well, if you read the "every number is interesting" "proof", this clearly
doesn't capture the proof's criteria of interestingness.

I see it as analogous to Berry's paradox - the proof isn't "wrong" per se, but
the relevant notion of interestingness is not well defined

~~~
gcr
What's not captured by this criteria?

You can express whatever idea of "interestingness" you like in this framework
by finding a predicate that expresses it.

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rosstex
I’m feeling more excited about this year already!

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tpogge
2017 is the sum of five distinct cubes TWICE OVER: 2017 = 10^3 + 9^3 + 6^3 +
4^3 + 2^3. 2017 = 12^3 + 6^3 + 4^3 + 2^3 + 1^3. As a bonus, it's also the sum
of EIGHT DISTINCT CUBES: 2017 = 9^3 + 8^3 + 7^3 + 6^3 + 5^3 + 4^3 + 3^3 + 1^3.

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duomono4
* (loop for i from 1 to 15 do (loop for i2 from i to 15 do (loop for i3 from i2 to 15 do (loop for i4 from i3 to 15 do (loop for i5 from i4 to 15 do (if (= (+ (* i i i) (* i2 i2 i2) (* i3 i3 i3) (* i4 i4 i4) (* i5 i5 i5)) 2017) (print (list i i2 i3 i4 i5))))))))

(1 2 2 10 10) (1 2 4 6 12) (2 4 6 9 10) NIL

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jmokland
44^2+9^2

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peter303
Definately a case of OCD. I am guilty of it too while interpreting license
plate numbers in boring traffic.

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leo424y
Pretty Cool!

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warent
Nice! Although I think it's amusing that they said "odd primes" as if there's
any even primes

~~~
otalp
2 is a prime number...

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warent
True!

~~~
vlasev
The oddest one of them all

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amag
Yup, it's so odd it's even!

