What I always wonder though, is 2017 really a special number or can you fit things like this to every number?
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...
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.
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?
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.
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?
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
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...
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.
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?
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…
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.
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?
Proved only for non-negative numbers! :)
Use the absolute value, maybe? You might have duplicates, but both n and -n can be special :)