5 ms·

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

Houshalter 10y ago
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&go=Search http://oeis.org/search?q=seq%3A2016&sort=&language=english&g...
- gcr 10y ago
  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?
  - mikeash 10y ago
    Can this proof be adapted for the reals, or is it only the case that every integer is special?
    - wbhart 10y ago
      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.
      - RangerScience 10y ago
        > On the other, other hand 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. http://catb.org/jargon/html/O/on-the-gripping-hand.html http://catb.org/jargon/html/O/on-the-gripping-hand.html
      - mikeash 10y ago
        I would argue that is is a proof of something, namely that you can't have a rigorous mathematical definition of specialness.
    - dwe3000 10y ago
      Outside the set of integers, the only reals left are either only fractionally interesting or crazy </end stupid pun>
    - Someone 10y ago
      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 {x | ∃ n ∈ ℕ : 10^n = 1} = {1, 0.1, 0.01, 0.001, 0.0001, 0.00001, …} doesn’t have a smallest number. 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.
  - smitherfield 10y ago
    Proved only for non-negative numbers! :)
    - mrkgnao 10y ago
      Use the absolute value, maybe? You might have duplicates, but both n and -n can be special :)
  - Houshalter 10y ago
    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 https://www.youtube.com/watch?v=_YysNM2JoFo
    - yodon 10y ago
      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. Oh, and happy new year's! [0] https://news.ycombinator.com/item?id=13281787 https://news.ycombinator.com/item?id=13281787
    - vog 10y ago
      > 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.)
    - Someone 10y ago
      ”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 http://oeis.org/A000027: ”The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.”
      - Houshalter 10y ago
  - aisofteng 10y ago
    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 10y ago
      > 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 10y ago
        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.
    - Someone 10y ago
      ”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?
  - inimino 10y ago
    What about the second such number? (Waiting for the new OEIS sequence of uninteresting numbers.)
- nocoder 10y ago
  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.
  - Houshalter 10y ago
    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.
tjwei 10y ago
Reminds me of "Interesting number paradox" :p This can be considered as a proof that every number is interesting. IMHO, being a prime number might give 2017 some advantages, and 2017 might be a slightly more interesting than most of prime numbers.
olalonde 10y ago
It turns out all natural numbers are special: https://en.wikipedia.org/wiki/Interesting_number_paradox https://en.wikipedia.org/wiki/Interesting_number_paradox

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&go=Search http://oeis.org/search?q=seq%3A2016&sort=&language=english&g...

gcr 10y ago
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?
- mikeash 10y ago
  Can this proof be adapted for the reals, or is it only the case that every integer is special?
  - wbhart 10y ago
    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.
    - RangerScience 10y ago
      > On the other, other hand 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. http://catb.org/jargon/html/O/on-the-gripping-hand.html http://catb.org/jargon/html/O/on-the-gripping-hand.html
    - mikeash 10y ago
      I would argue that is is a proof of something, namely that you can't have a rigorous mathematical definition of specialness.
  - dwe3000 10y ago
    Outside the set of integers, the only reals left are either only fractionally interesting or crazy </end stupid pun>
  - Someone 10y ago
    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 {x | ∃ n ∈ ℕ : 10^n = 1} = {1, 0.1, 0.01, 0.001, 0.0001, 0.00001, …} doesn’t have a smallest number. 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.
- smitherfield 10y ago
  Proved only for non-negative numbers! :)
  - mrkgnao 10y ago
    Use the absolute value, maybe? You might have duplicates, but both n and -n can be special :)
- Houshalter 10y ago
  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 https://www.youtube.com/watch?v=_YysNM2JoFo
  - yodon 10y ago
    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. Oh, and happy new year's! [0] https://news.ycombinator.com/item?id=13281787 https://news.ycombinator.com/item?id=13281787
  - vog 10y ago
    > 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.)
  - Someone 10y ago
    ”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 http://oeis.org/A000027: ”The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.”
    - Houshalter 10y ago
- aisofteng 10y ago
  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 10y ago
    > 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 10y ago
      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.
  - Someone 10y ago
    ”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?
- inimino 10y ago
  What about the second such number? (Waiting for the new OEIS sequence of uninteresting numbers.)
nocoder 10y ago
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.
- Houshalter 10y ago
  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.

tjwei 10y ago

Reminds me of "Interesting number paradox" :p This can be considered as a proof that every number is interesting. IMHO, being a prime number might give 2017 some advantages, and 2017 might be a slightly more interesting than most of prime numbers.

olalonde 10y ago

It turns out all natural numbers are special: https://en.wikipedia.org/wiki/Interesting_number_paradox https://en.wikipedia.org/wiki/Interesting_number_paradox