#number

[phonenumberwizard]

Telephone Codes for Australian States: A Complete Guide

Introduction

[a-number-a-day]

Today's number is 145

Somehow, I've done posts for 144 and 146, but not 145 . . .

In number theory, a happy number is a number that eventually reaches 1 when the number is replaced by the sum of the square of each digit.

For example, 13 is a happy number because

13 -> 1^2 + 3^2

= 10 -> 1^2 + 0^2

= 1

1 is itself a happy number because 1^2 = 1.

But there are numbers that are not happy. For example, 4 is an unhappy number because it gets stuck in an infinite loop (known as the Melancoil, whose name was popularized by Matt Parker):

4 -> 4^2

= 16 -> 1^2 + 6^2

= 37 -> 3^2 + 7^2

= 58 -> 5^2 + 8^2

= 89 -> 8^2 + 9^2

= 145 -> 1^2 + 4^2 + 5^2

= 42 -> 4^2 + 2^2

= 20 -> 2^2 + 0^2

= 4

145 is an interesting number because it is the only 3 digit number in the Melancoil.

All Numbers are Happy or Unhappy

I figured it was worth talking about how all numbers are happy or unhappy, and thus all numbers will eventually reach 1 or 145.

First note that large numbers always decrease. Suppose a number n has d digits. The largest possible value of f(n) occurs when every digit is 9, so f(n) <= 81d.

You can then compare 81d with the smallest d-digit number, 10^{d-1}.

For d >= 4, 81d <= 10^{d-1}

For example,

d = 4 : 324 < 1000 d = 5 : 405 < 10000

and it only gets more true as d grows.

Therefore, whenever n has at least 4 digits, f(n) < n. Thus, every sufficiently large number with at least four digits strictly decreases after one step.

If n has at most 3 digits, then

f(n) <= 3 * 81 = 243. Thus after at most a few iterations, every orbit enters the set

{1, 2, . . . , 243}

Once you're inside this set, every future value stays inside it because the sum of squares of three digits is still at most 243. Then you can use the fact that finite sets force cycles. If you keep iterating forever, eventually some value must repeat (this is essentially the pigeonhole principle).

Now the problem is finite: check the numbers 1, . . . , 243

A computer search shows there are exactly two cycles:

1 -> 1

4 -> 16 -> 37 -> 58 -> 89 -> 145 -> 42 -> 20 -> 4

Since every orbit must eventually enter a cycle, and these are the only two cycles, every positive integer eventually reaches either 1 or the Melancoil.

Factorion

As a final interesting fact, 145 is also a factorion, meaning that

145 = 1! + 4! + 5! = 1 + 24 + 120

It's one of only 4 factorions in base 10 (the others being 1, 2, and 40585).

Mathematicians have proven mathematically that no factorion can exist beyond seven digits. Any 8-digit number is at least 10,000,000, but the largest possible sum of the factorials of eight digits is only

8 * 9! = 2,903,040

Therefore, no 8-digit number can be a factorion. The same argument works even more strongly for numbers with more digits, so every factorion must have at most 7 digits.

Conclusion

So overall, 145 is an interesting number. It's the only 3-digit number in the Melancoil, but it's also a factorion, which are incredibly rare.

[bonguri]

20260509 Meitetsu Hiromi line 5 by Bong Grit Via Flickr: 何らかの意味のある数字なんだろうけど、わたしは解読できない。 Photo taken at Mitakeguchi station, Mitake town, Kani district, Gifu pref.

[frog-army]

did anyone make number headcannons instead of paying attention to math?

ima talk bout mine for those interested

so 7 and 3 are dating (started at 7 x 3= 21) but 7 is kind of closed off in the relationship which causes them to break up (time wise this is when 7 x 6= 42) but they get together again once they reach ( 7 x 8) and finally work it out and 7 proposes to 3 at the same time as 3 proposing to 7 (9 x 3 = 27) and they finally get married once they reach 70 and 30

4 is in love with 6 and 6 is toxic and only interacts with 4 when they need to get something from them (4 x 6= 24) 8 notices this and has a crush on 4 so tries to help them because 8 can't bear to see 4 getting used like this (4 x 8= 32). Because of this 4 starts developing feelings for 8 but thinks it's just admiration so keeps trying to date 6. (this is as far this has gone)

7 and 6 are mortal enemies and keep trying to kill the other. 6 once helped 7 but it was out of spite and to show as if 6 was better then them (7 x 6 = 42). this deeply angered 7 so they focused on revenge but 8 opened their eyes to what really mattered, aka 3. (7 x 8 = 56)

1 is desperately in love with 2 who just sees them as that annoying dude, 2 however is in love with 8 and keeps trying to reach them to confess. 1 is aware of this and decided to push their love away to help 2 because 1 believes seeing 2 happy is more important then their happiness. once (2 x 8 = 16) 8 rejects 2 and they finally realize how important 1 was but 1 had already left realizing that they shouldn't spend all their time on someone who doesn't love them.

9 is closed off and refuses to open up to anyone but loves playing match maker, their favorite ship is 7 and 3. 9 is also demi but because they never got close to anyone they never realized until (1 x 9= 9) they meet 1 and develop feelings. having never had them before 9 doesn't know what to do so is hiding their crush on 1.

10 is the chill aro/ace dude who actually remembers the quadratic formula's name. They are friends after all who enjoy watching the others like it's a live action movie.

the quadratic formula is that one guy everyone knows but almost no one can remember the name of but everyone knows. Think the man in the tan jacket from wtnv.

5 is a wild scientist that makes very dangerous experiments. One time 5 cloned themself and now they work together to make wild experiments with no one to stop them.

x took huge inspo of themselves from the letter x when they were a teenager. When they matured they wanted a more notable appearance so changed to · but now . is trying to sue them. Since they couldn't handle the lawsuit they changed themself to * out of spit to show how cooler they were to .

= and ≠ are twins, but, ≠ will never leave their emo phase.

+ and - have an enemies to lovers relationship. We're talking they literally were so close to killing each other that + has a scar from it that - did in the shape of themselves to prove something.

anyway is anyone surprised that i don't really have math skills now? like i was making these headcannons instead of actually doing school ✨procrastination✨

[q--ult]

number 7 sucks fuck number 7 ugly aah number

[dougfort]

[phonenumberwizard]

All Countries Calling Code: A Complete Guide to International Dialing

Introduction

[a-number-a-day]

Today's number is 142857

142857 is one of the most famous "curious numbers" in mathematics. It appears as the repeating digits of 1/7, and it has several surprising properties that make it worth remembering.

Cyclic Number

This number is the best known cyclic number in base 10. It is the six repeating digits of 1/7 (0.\overline{142857}) If you multiply 142857 by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, and 6/7 respectively:

1 * 142,857 = 142,857 -> 1/7 = 0.\overline{142857}

2 * 142,857 = 285,714 -> 2/7 = 0.\overline{285714}

3 * 142,857 = 428,571 -> 3/7 = 0.\overline{428571}

4 * 142,857 = 571,428 -> 4/7 = 0.\overline{571428}

5 * 142,857 = 714,285 -> 5/7 = 0.\overline{714285}

6 * 142,857 = 857,142 -> 6/7 = 0.\overline{857142}

7 * 142,857 = 999,999 -> 7/7 = 0.\overline{999999} = 1

If multiplying by an integer greater than 7, there's a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits to the remaining digits and repeating this process until only 6 digits are left, it will result in a cyclic permutation of 142857. For example, 815 * 142,857 = 116428455

Take the last 6 digits (428,455), and add them to the remaining digits:

116 + 428,455 = 428,571, which is a cyclic permutation.

This happens because 10^6 is congruent to 1 (mod 142857). Because of that, replacing a block of digits by the sum of its six-digit chunks preserves the remainder modulo 142857.

Mental Division

I've actually used this trick before when I needed to divide by 7 quickly. If you have some natural number, it's fairly trivial to find the integer part. Once you have it, you can then divide the remainder by 7 and consider 142857. If your remainder is n, you look at n/7 in the table above.

For example, to compute 103 / 7, first note that 14 * 7 is 98, leaving a remainder of 5. Then note that 5 / 7 = 0.\overline{714285}.

So the 103 / 7 = 14.\overline{714285}.

Kaprekar Number

142857 also appears in a completely different corner of recreational mathematics.

A p-Kaprekar number is a number whose square can be split into two parts (whose second part is p-digits long) and add to the original number. For example, 45 is a 2-Kaprekar number because 45^2 = 2025, and 20 + 25 = 45.

142857 is a 6-Kaprekar number since 142857^2 = 20,408,122,449, and 20,408 + 122,449 = 142,857.

Conclusion

Overall, 142857 is a pretty cool number. So the next time you get mugged and are asked to find 7352 / 7 to at least six decimal points of accuracy, you won't be in trouble.