The Reciprocals of Primes - Numberphile

Matt Parker explores the work of William Shanks - and boots up the ShanksBot.
More links & stuff in full description below ↓↓↓

NUMBERPHILE

Videos by James Hennessy and Brady Haran

[pXnTilde]

Its so crazy that anyone with some math and programming experience can do in an afternoon what took him years of his life. But those tools allow us to go so much further.

[Rational_thinker_]

in 1982, i was working as a lab assistant and got access to a Hewlit Packard calculator the size of 3 loaves of bread that was programable in BASIC. After learning the language i wanted to write a program that would keep the device busy over night. Everyone told me that it was so fast i would never be able to do it. I wrote a program that simply diveded intengers by each other from an array of 1000 x1000 and reported the number of digits before a repeat. After the first night it ran out of paper reporting so I altered it to only tell me when it got a result over 1000 digits. I became facinated with this and kept increasing the array size and only looking at larger and larger numbers. I soon had the poor machine running none stop over three day weekends without finishing.

Some of the PhDs in the lab started to take notice and asked me to program things for them as they thought I was some kind of math/programmer genius. In fact I had no training past highschool trig. But with the the help of few references books, I hade the poor overworked gloified calculator running regressions and fast fourier transformations that before had to have a FORTRAN programmer write, punch cards and buy time on a mainframe computer.

Great fun, imagine if Shanks would have had a programmable calculator.

[thenakedsingularity]

You really got to respect this William Shanks guy. He also did the longest calculation of Pi before computers.

[train_blabber]

There was a mistake in Matt's 1/23 division 😳

When the number at the bottom is 2, he carries two zeros, so there should be 08 in the top, not just 8.

[EchosTackyTiki]

12:30 I think the fact that he just did it because he wanted to, not because it was useful, but just because he thought it was fun, is the most endearing part of his story. I like little historical footnotes like that. I think sometimes they make all the difference.

[DMCmph]

After watching Matt Parker perform impressive math for YEARS, I'll gladly help him with the long division with which he seems to struggle, lol.

[Bleighckques]

Can't believe a Numberphile video came out related to something that I've actually been doing myself. Not to this scale, but I've been trying to do these in my head with the smaller primes every once in awhile if I can't sleep (I think I'm up to 61 or 67). I use the same method of long division, but I don't keep track of the actual value of the reciprocal because I lose track, I just count how many steps before I reach 1 again.
A couple interesting things I've found:
- The point about doubling or halving the solutions Shanks had isn't due to how he's acquiring the answers but rather due to a property they all share. Not only do all the answers fall into the range of 1 to n-1, but they are all factors of n-1. For example, 13 has 6 digits that repeat, 13-1 is 12, of which 6 is a factor; 53 has 13 digits that repeat, 53-1 is 52, of which 13 is a factor, etc. So it would be common to have a mistake that leads you to count past the halfway point and simply conclude that the answer must be n-1 because it is the only remaining factor after reaching (n-1)/2, leading to later having to halve the answer to correct it.
- The numbers that have n-1 digits in their answers all have just one string that repeats for every fraction in the range 1/n to (n-1)/n. 7 for example, 1/7 = 0.142857, 2/7 = 0.285714, 3/7 = 0.428571, 4/7 = 0.571428, 5/7 = 0.714285, 6/7 = 0.857142. However, the numbers with answers that are some factor (n-1)/x will have x different sequences of numbers. 13 for example, 1/13, 3/13, 4/13, 9/13, 10/13, and 12/13 all repeat 076923 with different starting points, but 2/13, 5/13, 6/13, 7/13, 8/13, and 11/13 all repeat 153846 instead.

[pectenmaximus]

“It’ll seem random for 5001 digits, and then you’re like, ‘this is familiar’”

Yeah, happens all the time

[csolus]

You forgot to mention this nice property: if you take the repeating "number" (e.g. 142857 for n=7), you get a number N with a very nice property: every multiple of it is a circular permutation (e.g. 2*142857 = 285714), which also extends to products by any number, assuming that you chop the result in chunks of size n-1 and add them up. For instance 142857^2 =20408122449, and 122449+20408=142857 ;-) ;-) It's quite obvious given the origin of this number, but the property is really cool.

[pickleballer]

"His only practical application for what he's done, is you can use it to find another thing with no practical application." The amateur mathematician's creed! I worked out the algorithms for getting all of the irreducible Pythagorean sets, the sums of squares and cubes up to n, and worked on Fermat's Last Theorem for a long time just because I was curious. A friend of mine asked me over and over why I did that when I could have just looked it up (I never imagined that I was the first to do any of that at all.) The only answer I had was. "It was fun."

[irakyl]

Brady's question of "will all primes hit an infinite loop like this?" has an easier answer. All reciprocals are periodic, because if they didn't have a period, they would be irrational, and they are clearly defined as a ratio of 1 and the prime.

[thomaswagner]

Fascination - brings back memories of high school back in 1968. I wrote a Fortran program to compute how long of a repeating sequence that the inverse of an odd number was. My program worked correctly and I had fun using it, and showing the results to my high school math teachers.

[charlottedarroch]

The way Shanks likely did it was by calculating the multiplicative order of 10 modulo the given prime p. This order always divides p-1 and there are a variety of tricks to make modular arithmetic by hand much easier. And so, for example, to show that the order of 10 mod 60013 is 5001, one would need to show that 10^5001 = 1 mod 60013, but that 10^d =/= 1 mod 60013 for any proper factor d|5001. So one would need only check the factors 1, 3 and 1667.

[Gribbo]

Loved the lesson in long division by hand. This arithmetic process we learned at the age of about 6 or 7 back in my day (70 years ago!). So it seems a bit strange to see a serious mathematician like Matt working his way through it step by step (doesn't everybody know how to do this?). But truth be told, I tried to do a long division by hand while back and was shocked to find I'd forgotten how to do it after 50 or so years of calculators! Thanks for the refresher :)

[saetmusic]

My brother spent decades completely obsessed with Prime Numbers. Shortly before he passed away, he sent me a thick packet of his research. He believed he had discovered "The Codification of the Primes" and "The Periodic Table of the Primes" as Mentioned in Santoy's book, "The Music of the Primes" There are handwritten pages of notes and then there are dozens of pages just filled with numbers. I can't make anything of it, but is it something you might be interested in seeing?

[priyanshupaswan]

Getting numberphile, stand up maths and objectivity notification at the same time.. gonna have a great time

[Paul-sjdb]

So, Schenk ran a boarding school. Maybe he crowdsourced the process by getting the children to do a prime, as punishment or just normal work. With a few children doing the same prime he could have some error correction as long as they didn't copy off each other.

The strangest part of this was the relative lack of curiousity of how he did it.

[igrim]

10:24, Matt: (reads line in book) 60017 is as bad as it gets. It hits all 60016 possible minuends.
Brody: (zooms in on book to show the very next line, 60029 hits all 60028 of its possible minuends)

[peterpike]

The way that digits repeat depends on the base you're using too. For example, 1/7 = 0.142857... in base 10, but in base 6, the same value would be 1/11, and it would be 0.0505050505... And most fun would be base 8, where 1/7 Just like in base 10, 1/9 = and for the same reason: 1/[base-1] is always going to equal So if you want to know what the reciprocal to any particular number is without having to calculate it out, now you know for at least the base that's one more than the value you're looking for.

[kektagonb]

I don’t know what I was expecting coming to watch this video, but I definitely didn’t expect to finally truly understand long division. Something about how you demonstrate it so simply made it finally click.