[math-fun] Lexicographic ordering of powers of two ...
Here's a small thing that a friend of mine discovered: https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful. Thanks.
Something similar will work for the powers of 3/2 in base two; since (3/2)^12 is close to a power of 2, the powers of 3/2, written in binary and then sorted into lex order, are close to the powers of the twelfth root of 2. This is the basis of equal tempering in Western music. Jim Propp On Fri, Jul 26, 2019 at 9:12 AM Colin Wright <maths@solipsys.co.uk> wrote:
Here's a small thing that a friend of mine discovered:
https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf
Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful.
Thanks.
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this works because 2^10 is roughly 10^3, right? - Cris
On Jul 26, 2019, at 7:12 AM, Colin Wright <maths@solipsys.co.uk> wrote:
Here's a small thing that a friend of mine discovered:
https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf
Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful.
Thanks.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Right, just like the other one works because 3^12 is close to 2^19. Jim On Sun, Jul 28, 2019 at 2:58 PM Cris Moore <moore@santafe.edu> wrote:
this works because 2^10 is roughly 10^3, right? - Cris
On Jul 26, 2019, at 7:12 AM, Colin Wright <maths@solipsys.co.uk> wrote:
Here's a small thing that a friend of mine discovered:
https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf
Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful.
Thanks.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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What your friend observed is essentially a consequence of Weyl’s theorem: Let alpha be irrational, then the sequence n alpha mod 1 is dense in the interval [0,1]. Since log 2/log 10 is irrational, this applies. Victor On Fri, Jul 26, 2019 at 09:12 Colin Wright <maths@solipsys.co.uk> wrote:
Here's a small thing that a friend of mine discovered:
https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf
Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful.
Thanks.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hello , is it what you mean ? See sequences : http://oeis.org/A158624 and : https://oeis.org/A023415 These are the limit values of 5^n and 2^n backward. Simon Plouffe and sequence Le dim. 28 juil. 2019 à 22:58, Victor Miller <victorsmiller@gmail.com> a écrit :
What your friend observed is essentially a consequence of Weyl’s theorem: Let alpha be irrational, then the sequence n alpha mod 1 is dense in the interval [0,1]. Since log 2/log 10 is irrational, this applies.
Victor
On Fri, Jul 26, 2019 at 09:12 Colin Wright <maths@solipsys.co.uk> wrote:
Here's a small thing that a friend of mine discovered:
https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf
Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful.
Thanks.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I should explain a little more. If, as your friend did on his web page, puts a decimal point after the first digit of 2^n, and then looks at the ordering (i.e. sorts them), this is the same order as if you sorted the log base 10 of 2^n modulo 1 (i.e. subtract the integral part). The Weyl equidistribution theorem says that for any interval [a,b] in [0,1] that the ratio of the number of n alpha mod 1, for n=1, ..., N that land in that interval to N converges to b-a as N --> infinity. Since we're taking logarithms base 10, we're comparing that with the sequence n/10 mod 1. For a quanitative version look at the Erdos-Turan inequality: https://en.wikipedia.org/wiki/Erdős–Turán_inequality On Sun, Jul 28, 2019 at 5:08 PM Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello ,
is it what you mean ? See sequences : http://oeis.org/A158624 and : https://oeis.org/A023415 These are the limit values of 5^n and 2^n backward.
Simon Plouffe
and sequence
Le dim. 28 juil. 2019 à 22:58, Victor Miller <victorsmiller@gmail.com> a écrit :
What your friend observed is essentially a consequence of Weyl’s theorem: Let alpha be irrational, then the sequence n alpha mod 1 is dense in the interval [0,1]. Since log 2/log 10 is irrational, this applies.
Victor
On Fri, Jul 26, 2019 at 09:12 Colin Wright <maths@solipsys.co.uk> wrote:
Here's a small thing that a friend of mine discovered:
https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf
Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful.
Thanks.
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participants (5)
-
Colin Wright -
Cris Moore -
James Propp -
Simon Plouffe -
Victor Miller