2017 is not just another prime number
weijr-note.blogspot.com
weijr-note.blogspot.com
This gives no results in SageMath...
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.]
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, 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...)
If that is true, it is indeed common that 5 cubes is enough (since 4 almost always would be sufficient)
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.
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?
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
https://www.youtube.com/watch?v=z6jMU-AwX34
(Some repeats, but plenty of non-prime facts as well (plus Matt's excellent dry humor))
http://oeis.org/search?q=seq%3A2016&sort=&language=english&g...
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?
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.
{x | ∃ n ∈ ℕ : 10^n = 1}
=
{1, 0.1, 0.01, 0.001, 0.0001, 0.00001, …}
It is impossible to adapt this proof because the set of reals is uncountable. It is doable for the rationals, though, as they _are_ countable.
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
Oh, and happy new year's!
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.)
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.”
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.
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
You can express whatever idea of "interestingness" you like in this framework by finding a predicate that expresses it.
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?
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.
(Waiting for the new OEIS sequence of uninteresting numbers.)
IMHO, being a prime number might give 2017 some advantages, and 2017 might be a slightly more interesting than most of prime numbers.
(1 2 2 10 10) (1 2 4 6 12) (2 4 6 9 10) NIL
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.