mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2020-08-04, 14:50   #12
xilman
Bamboozled!
 
xilman's Avatar
 
"π’‰Ίπ’ŒŒπ’‡·π’†·π’€­"
May 2003
Down not across

101000000011102 Posts
Default

Quote:
Originally Posted by storm5510 View Post
All of this seems to make something very simple into something very complex. It is not. If you take them in pairs, (one of each type), the count will be the same for odds and evens, if the counting stops on an even.

(1,2)(3,4)(5,6)(7,8) and so on.
And if it does not?

You have provided a proof of theorem that there is one more odd natural number than there are evens.
xilman is offline   Reply With Quote
Old 2020-08-04, 15:52   #13
JeppeSN
 
JeppeSN's Avatar
 
"Jeppe"
Jan 2016
Denmark

2338 Posts
Default

You consider d(N) as the difference between the number of odd natural numbers under N and the number of even natural numbers under N. Then d(N) does not converge for N tending to infinity, in the usual sense. Therefore you consider the sequence of arithmetic means of all d(i) for i up to N, and now it converges. The limit, if everything goes the way I think, reveals what definition of "natural number" you have. /JeppeSN
JeppeSN is offline   Reply With Quote
Old 2020-08-04, 16:14   #14
xilman
Bamboozled!
 
xilman's Avatar
 
"π’‰Ίπ’ŒŒπ’‡·π’†·π’€­"
May 2003
Down not across

2·3·1,709 Posts
Default

Quote:
Originally Posted by JeppeSN View Post
You consider d(N) as the difference between the number of odd natural numbers under N and the number of even natural numbers under N. Then d(N) does not converge for N tending to infinity, in the usual sense. Therefore you consider the sequence of arithmetic means of all d(i) for i up to N, and now it converges. The limit, if everything goes the way I think, reveals what definition of "natural number" you have. /JeppeSN
This thread is in "Miscellaneous Math" ...

Toto, I've a feeling we're not in Kansas any more.
xilman is offline   Reply With Quote
Old 2020-08-04, 17:44   #15
storm5510
Random Account
 
storm5510's Avatar
 
Aug 2009
U.S.A.

3×7×73 Posts
Default

Quote:
Originally Posted by xilman View Post
And if it does not?

You have provided a proof of theorem that there is one more odd natural number than there are evens.

Alright, I will take your word for it, and not belabor the issue. One question though: Where does the extra odd number reside?
storm5510 is offline   Reply With Quote
Old 2020-08-04, 17:59   #16
xilman
Bamboozled!
 
xilman's Avatar
 
"π’‰Ίπ’ŒŒπ’‡·π’†·π’€­"
May 2003
Down not across

2×3×1,709 Posts
Default

Quote:
Originally Posted by storm5510 View Post
Alright, I will take your word for it, and not belabor the issue. One question though: Where does the extra odd number reside?
You have already told us.

Quote:
Originally Posted by storm5510 View Post
All of this seems to make something very simple into something very complex. It is not. If you take them in pairs, (one of each type), the count will be the same for odds and evens, if the counting stops on an even.

(1,2)(3,4)(5,6)(7,8) and so on.
"if the counting stops on an even": if it does not, the extra odd number is that one which does not have an even counterpart.
xilman is offline   Reply With Quote
Old 2020-08-06, 10:42   #17
Nick
 
Nick's Avatar
 
Dec 2012
The Netherlands

25×32×5 Posts
Default

Quote:
Originally Posted by retina View Post
The asymmetry that lies behind your initial post stems from the fact that the exponents in ordinary prime factorization are restricted to non-negative integers.
If we work with fractions (or, more generally, fractional ideals in any Dedekind domain) then this problem disappears.
Nick is offline   Reply With Quote
Old 2020-08-06, 10:57   #18
retina
Undefined
 
retina's Avatar
 
"The unspeakable one"
Jun 2006
My evil lair

32·72·13 Posts
Default

Quote:
Originally Posted by Nick View Post
The asymmetry that lies behind your initial post stems from the fact that the exponents in ordinary prime factorization are restricted to non-negative integers.
If we work with fractions (or, more generally, fractional ideals in any Dedekind domain) then this problem disappears.
I'll have to think about that.
retina is offline   Reply With Quote
Old 2020-08-08, 16:23   #19
kriesel
 
kriesel's Avatar
 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

24·277 Posts
Default

Quote:
Originally Posted by retina View Post
4, 8, 16 ,32, 64
(in response to:)
Originally Posted by kriesel
Quote:
Do tell. Give a list of five of them that are not immediately preceded by an odd number each.
4, 8, 16 ,32, 64 are each immediately preceded by 3, 7, 15, 31, 63, respectively; Mersenne numbers 2n-1 precede powers 2n.

Last fiddled with by kriesel on 2020-08-08 at 16:26
kriesel is offline   Reply With Quote
Old 2020-08-08, 16:40   #20
kriesel
 
kriesel's Avatar
 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

443210 Posts
Default

For any given binary integer word size, of a positive whole number of bits n>0, there are exactly as many evens as odds.
https://en.wikipedia.org/wiki/Two%27s_complement. The difference being zero for any n, the slope of the difference function is zero, and the difference in the limit at n=infinity is also a difference of zero. It does not change if considering unsigned integers, since 0 to 2n-1 is also comprised of exactly as many evens as odds.

Last fiddled with by kriesel on 2020-08-08 at 16:44
kriesel is offline   Reply With Quote
Old 2020-08-08, 20:10   #21
kruoli
 
kruoli's Avatar
 
"Oliver"
Sep 2017
Porta Westfalica, DE

13316 Posts
Default

Quote:
Originally Posted by kriesel View Post
...preceded...
We are not talking about proceeding like 1, 2, 3. Someone gave a specific rule to follow, so this person set how to "proceed". In this case, all of retina's examples are perfectly valid.
kruoli is offline   Reply With Quote
Old 2020-08-08, 23:11   #22
kriesel
 
kriesel's Avatar
 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

10001010100002 Posts
Default

Quote:
Originally Posted by retina View Post
Every odd number is half of an even number.

Some even numbers are not double an odd number.

Therefore there are more even numbers than odd numbers.
The original post does not indicate a number system explicitly.
Would that be natural numbers, all integers, all rationals?
Some of the online definitions for odd number are imprecise.

https://duckduckgo.com/?t=ffnt&q=odd...&ia=definition

Is x= 1 / 2 odd or even?
In what bases?
At https://www.mathsisfun.com/numbers/even-odd.html,

"Any integer that can be divided exactly by 2 is an even number."
x=1/2 = exactly 0.5 (base 10) =0.3 (base 6) is even, but x=1/2 = 0.3.. (base 7) so one is both even and odd, by that definition, depending on the number base in which the calculation is attempted. (They do continue on to give examples of odd numbers that imply integer computation only, no reals.)

https://www.rapidtables.com/calc/mat...alculator.html
is a selectable base calculator.

Last fiddled with by kriesel on 2020-08-08 at 23:12
kriesel is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Numbers sum of two cubes and product of two numbers of the form 6^j+7^k enzocreti enzocreti 2 2020-02-16 03:24
Devaraj numbers which act like Carmichael numbers devarajkandadai Number Theory Discussion Group 1 2018-07-30 03:44
Carmichael numbers and Devaraj numbers devarajkandadai Number Theory Discussion Group 0 2017-07-09 05:07
6 digit numbers and the mersenne numbers henryzz Math 2 2008-04-29 02:05
LLT numbers, linkd with Mersenne and Fermat numbers T.Rex Math 4 2005-05-07 08:25

All times are UTC. The time now is 01:58.

Sat Sep 26 01:58:18 UTC 2020 up 15 days, 23:09, 0 users, load averages: 2.06, 1.84, 1.80

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.