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:
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...)
If that is true, it is indeed common that 5 cubes is enough (since 4 almost always would be sufficient)
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.
> 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.
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.
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.
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?
Well the first number that doesn't appear in OEIS is 18159. That make it interesting to me, but apparently this number isn't interesting to mathematicians.
Of course since OEIS is finite, there will always be numbers which don't make it. What's really interesting is that some numbers are disproportionately underrepresented. They appear much less often than other numbers of the same size. And if you plot each number by the number of times it occurs, it makes a really interesting pattern: https://www.youtube.com/watch?v=_YysNM2JoFo
Fascinating video - the distribution seems likely related to the HN post from two days ago on multiple random processes being sufficient to generate Zipf's law distributions [0] (given the observation a couple posts above that the first number not in the list is 18159 and the steady average decrease in frequency with value there isn't really much difference between rank-ordering and value ordering here). That exponent they find sure looks Zipf like.
”Of course since OEIS is finite, there will always be numbers which don't make it”.
18159 appears in OEIS, but its search interface isn’t perfect.
For example it appears in http://oeis.org/A000027: ”The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.”
Yes 18159 does fit the definition of a natural number. But OEIS only contains the first few terms of every sequence. So I'm technically correct that it isn't in OEIS. And I would argue that being the 10k+ term in a sequence doesn't qualify as interesting.
> Well the first number that doesn't appear in OEIS is 18159. That make it interesting to me, but apparently this number isn't interesting to mathematicians.
I have a diploma in math (equiv. to master of science in math) and I'm interested in 18159. Does that count? (Moreover, I'm interested in 18159 for exactly the reason you stated.)
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.
”Specifically, a number n is interesting if there is some predicate P(x) which is true only for n”
That’s not that good a definition. Since the predicate “P(x) = x = 4578634986” is only true for x = 4578634986, would that imply that 4578634986 is interesting?
I think it was always intended to be sort of a joke. "Interesting" is subjective, but obviously any self-consistent criteria of interesting cannot allow something to be interesting by it's sheer uninterstingness.
But even if you're willing to accept the paradox, it's still a bullshit proof. It's one thing to say a number is interesting because it really is the first number you can't think of anything interesting to say about it. It's something else to say to say it's interesting because its the first number you can't find anything interesting about, not counting all the others that were considered "interesting" for the same reason.
It's not a proof of anything, so it can't be adapted to the reals. Specialness has no rigorous mathematical definition, and certainly the second non-special number is not special, even if the first is. But more mathematically speaking, given a standard real number, there is no "next" real number. Induction is one of the defining properties of the natural numbers and it is induction that allows one to construct proofs that show that some property holds for all natural numbers by showing that if it holds for n, then it holds for the successor of n. On the other hand, if you assume the axiom of choice, the real numbers are well-ordered. So if there is at least one non-special real number, then there is a least non-special real number with respect to this ordering, which makes it special (assuming there is something special about this particular well-ordering). Unfortunately, no one can exhibit a well-ordering of the reals, so it's a little hard to judge if any of them are "special". On the other, other hand, it is humans that seem to decide if numbers are special, and there are finitely many humans writing finitely many papers and other communications, which are thus countable. This implies that for there to be any chance of all real numbers being special, there must be uncountable families of special numbers. Once one starts thinking along these lines, one might recognise that all real numbers are special, because they are real (and not more generally complex). I can't personally think of any uncountable sets of numbers that are more special than the reals, except perhaps the transcendental reals. On the other hand, these are basically just all the real numbers that aren't special enough to be algebraic. So they aren't very special, and certainly not more special than the natural numbers.
You might like the alternative (and very nerdy) phrase "on the gripping hand" as an alternative. It (usually?) connotes a third option that invalidates the other two in some way, as you've done here.
No. ”Assume that not every number is mathematically interesting and let X be the first such number” doesn’t work as is because a set of reals need not contain a smallest number. For example, the set
Wouldnt a lot of instances of 2016 be simply because of its use as the year "2016" instead of anything mathematical? I found 38K instances of 2015, 42K of 2013 and 39K of 2014. I am not familiar with OEIS so just asking.
Yes you are right. That's a huge mistake on my part. I fixed it by using the prefix "seq:2017" so it only matches sequences that contain that number, not the metadata.
* (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))))))))
All throughout the talk there were statements like "Let p be an odd prime and…"
My friend asked, “what is an odd prime?”—thinking it must be special in some way. The answer back was: not 2.
