#142857

[math1089]

Mathematical Beauties of the Number 142587

There are some mysteries that the human mind will never penetrate. To convince ourselves we have only to cast a glance at tables of primes and we should perceive that there reigns neither order nor rule.Leonhard Euler Welcome to the blog Math1089 – Mathematics for All. I’m glad you came by. I wanted to let you know I appreciate your spending time here on the blog very much. I do appreciate your…

[positivelyprime]

My favourite fucked up math fact™ is the Sharkovskii theorem:

For any continuous function f: [a,b] -> [a,b], if there exists a periodic point of order 3 (i.e. f(f(f(x))) = x for some x in [a,b] and not f(x) = x or f²(x) = x), then there exists a periodic point of ANY order n.¹

Yes you read that right. If you can find a point of order 3 then you can be sure that there is a point of order 4, 5, or even 142857 in your interval. The assumption is so innocent but I cannot understate how ridiculous the result is.²

For a (relatively) self-contained proof, see this document (this downloads a pdf).

(footnotes under read more)

¹ The interval does not have to be closed, but it should be connected. (a,b), (a,b] and [a,b) all work.

² Technically the result is even stronger! The natural numbers admit a certain ordering called the Sharkovskii ordering which starts with the odd primes numbers 3 > 5 > 7 > ... , then doubles of primes odds, then quadruples of primes odds and so forth until you get no more primes odds, left, ending the ordering in 2³ > 2² > 2. Sharkovskii's theorem actually says that if you have a periodic point of order k, then you have periodic points of any order less than k in the Sharkovskii ordering.

Edit: corrected footnote 2

[hansen82]

whats your favorite number mine is 142857

🤷‍♂️ IDK.

[lvpwwfz10]

. . .

[sunview]

Lil pohpoh

[novossamples]

#NovosSamples: 142,857 by Sacred Geometry: Dagha & Mike P.

[math1089]

40 Quirky and Fascinating Facts about Mathematics

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. Leonhard Euler Glad you came by. I wanted to let you know I appreciate your spending time here at the blog very much. I do appreciate your taking time out of your busy schedule to check…

[melody-lines]

124875 || 142857 (Lyric video)

[unusualshrimp]

im sure everyone knows this already but its a bit Neat that when you divide a not-7's-multiple number by 7 you get a 142857 repeating in a circle

for example

22÷7 = 3.142857142857142857...

100÷7 = 14.2857142857142857...

do any other numbers do this or is 7 just a freak

[reed-right]

fun math fact since nobody asked: if you have a repeating decimal (for example, 0.1428571428571...), you can take the repeating part (142857) and divide it by the same number of digits but all 9's (142857 / 999999) to get the ratio of that repeating number. In many cases, you can then reduce the fraction (1 / 7).

also, because it's even somewhat relevant (and I'll take any opportunity to share this with people): for any number that is just a series of digits that are all 9 (for example, 999), the (repeating) decimal value of the reciprocal of the square of that number (1 / 999²) will list every single n-digit number (000 001 002 003 004 005...) except for the penultimate one (... 995 996 997 999 000 001...).

and of course, as with anything in math, these aren't coincidences; there's nothing special about the digit '9' here other than being the highest one in our base 10 (decimal) number writing system. this works in any base -- the same occurs in base 8 (octal) with the digit 7, in base 2 (binary) with the digit 1, in base 20 (vigesimal) with the digit 19, etc.

Reddit is such a perfect mixture of hauty arrogance and downright stupidity, sometimes. It's almost beautiful.

[grennsw]

This is a thing. Specifically a thing that I made. Hi, I am Gren. Just gonna info-dump about myself. An introduction, per se.

I hail from the UK. I play a lot of games, mostly Splatoon, Tetris and Mario. I am a speedrunner. I like lizards, as do 99% of this website. I stream on Twitch here where I play games and chat. I can play Geoguessr pretty well. I'm losing the will to live. I'm queer, whilst also being here. I made this to shitpost, to spread fun facts, and that's it. I have no idea how this site works yet. I like green-blue-purple colour schemes. I'm a big Pokémon fan. My favourite is Trapinch. I enjoy researching things that have very little meaning. And mathematics. My favourite/the best number is 142857. The perfect number is 6. And 28. And 496.

I'm not funny. I try my best. My favorite musician is Stromae. I'm learning to speak French. I'm a science nerd. I'm running out of things to talk about. Take care.

T'inquiète pas, ça va aller. C'est fini.

[thelreads]

142857 x 1 = 142857, 142857 x 2 = 285714, 142857 x 3 = 428571, 142857 x 4 = 571428, 142857 x 5 = 714285, 142857 x 6 = 857142, 142857 x 7 = 999999

[csdragon]

142857

Broke: 142857 is an important number to math, being the best known cyclical number in base 10, representing the cyclical nature of The Reaper’s game.

Woke: It’s prime factorization is 11, 13, 3, 3, 3, 37. LLLEEEEET

[bruxadepano]

1. I have insomnia since I was a small child

2. I can play some instruments, but don't know musical theory

3. I love art, but don't have the ability to make it

@hufflefuckersss, @thegirlnoonetexts, @saturnsbasement, @v-142857, @betwixttwoendlesswinters, @samblerambles, @skellybonesandtrees

If you get this, answer w/ three random facts about yourself and send it to the last seven blogs in your notifs. anon or not, doesn’t matter, let’s get to know the person behind the blog!

facts about me

i currently ID as grey/grayromantic

im asexual

i was born in june

blogs: @solacistic @samslin @tooningin (appeared twice in my notifs) @tiger-willow @thowawayuntilfurthernotice @friedballoonchild425

[twocubes]

Do you ever think about numbers outside of base ten and the patterns that are possible in these?

I mean I used to at some point... i think the last one i thought about was...

Like, ok, Fermat’s little theorem sez: given an integer a and a prime p that doesn’t divide a, aᵖ⁻¹ is congruent to 1 mod p, right?

In other words, aᵖ⁻¹-1 is a multiple of p. So like, if a is the base b of a number system, and, say, “x” is the symbol in base b that corresponds to b-1, then p always divides bᵖ⁻¹-1 = xxxxx...x (base b) ← p-1 digits long. 

But, obviously, in any base, 1=0.xxxxx... and thus 1/(xxxxx...x (base b) ← p-1 digits long) is just 0.00....01 repeating (with p-2 zeroes) and thus 1/p can be written as 0.(a repeating sequence of p-1 digits)

So, this is like how 1/7=0.142857... repeating. “142857″ is six digits long, as expected.

But you might ask, like, 1/7 has this other neat thing which is that all its multiples are cyclical permutations of the original:

1/7 = 0.142857... 2/7 = 0.285714... 3/7 = 0.428571... 4/7 = 0.571428... 5/7 = 0.714285... 6/7 = 0.857142...

and you could be like

“i wanna see that again in other bases, how do i look for it”

and, like, ok, what does “being a cyclical permutation of the original” mean?

If u think about it just a little, it means like, for each of these, you can, for each k=1...6, find an exponent n and some integer m such that (10^n)(1/7) = m + k/7, right.

In other words, it means that 10^n = k + 7m; in other words it means that there is n such that 10^n is congruent to k mod p for each k.

This all obviously generalizes to if you pick another base b instead of 10 and like, some other number p instead of 7.

So, equivalently, this means that we want the base b (modulo p) to generate the multiplicative group of ℤ/pℤ. So like, for example, we know 10 generates the multiplicative group of ℤ/7ℤ, so certainly 3 must as well. What do we see in base 3?

1/7 = 0.010212... 2/7 = 0.021201... 3/7 = 0.102120... 4/7 = 0.120102... 5/7 = 0.201021... 6/7 = 0.212010...

Nice! Ok, enough of 7, what’s a number that generates the multiplicative group for ℤ/13ℤ? Well, you can find that 2 does, but binary is a bit boring for this, so... 15?

1/13 = 0.124936DCA5B8... 2/13 = 0.24936DCA5B81... 3/13 = 0.36DCA5B81249... 4/13 = 0.4936DCA5B812... 5/13 = 0.5B8124936DCA... 6/13 = 0.6DCA5B812493... 7/13 = 0.8124936DCA5B... 8/13 = 0.936DCA5B8124... 9/13 = 0.A5B8124936DC... 10/13 = 0.B8124936DCA5... 11/13 = 0.CA5B8124936D... 12/13 = 0.DCA5B8124936...

Cool! In fact, the name for a number that generates the multiplicative group of ℤ/nℤ is a primitive root modulo n. There’s a nice big table of them on that wiki page.

...

anyways hopefully that answers your question.