Detailed list of attributions
When I first started this blog, I was 15 and homeschooled and had no sense of academic honesty. All I knew was that after reading books about math from big names such as John Conway and David Wells and thinking deeply about numbers, ideas came into my mind. These ideas were sometimes interesting enough that I thought it would do good to publish them, even though I didn't always remember where the thoughts came from or how much I had to read before I thought of the rest myself.
As I went on and gained more of a sense of how others would view my blog, I began to notice that there were very glaring omissions of any attribution from most of my posts. I was writing as if all my ideas were coming straight from my brain, which was true for some but not all of my posts. When I transitioned straight from homeschool into college in 2017, the most common reaction people had when hearing about my mathematical interests was "Did you come up with this yourself or find it somewhere?" I often could not answer because I had spent almost no time paying attention to where my ideas came from. Sometimes I had actually forgotten where I learned an advanced mathematical topic because I had known about it for so long, almost 10 years. Once I memorized the proof of an interesting phenomenon I saw, the conclusion seemed self-evident. Until then, as far as I was aware what I was learning was simply a refinement of my everyday thought and none of my posts needed attributions because everything that I didn't discover was assumed to be common mathematical knowledge.
Right before leaving for college I wrote the following disclaimer in my "About" page to try to demystify the origin of many of my ideas:
First, not everything I post here is one of my recent thoughts. Sometimes I decide to post an idea that I have had for up to ten years (I have 18). I don’t post every idea I have either; quite often, a post is so long that I started working on it only a few days after I finished the previous post.
As of this writing (August 17, 2017), I am not in college yet, though I will be going to New Mexico Tech in a few days, probably before anyone reads this.
Finally, I do not claim to be the first to discover any of the things I post here. (If I am though, it would be awesome.) Except where otherwise stated, the ideas in this blog were thought of by me, even though they sometimes build off of others’ ideas.
I still stand by everything I wrote. I wish I could have left it at that because it is not exactly fun for me to look up one of my "discoveries" and find that the exact same thing has been discovered by someone else in the past. But after years of hearing about the dangers of plagiarism and the importance of citing my sources, combined with the increasing popularity of my blog as I put it on my résumé to advertise my mathematical talent to potential employers, I think that it is time for a more comprehensive list of my sources of knowledge.
I'll go through my posts chronologically, recalling as best I can my knowledge for each of them.
Multiples of 37 and its sequel
To the best of my knowledge, I got the idea for this question by looking at a graph of the number of palindromes divisible by each prime, from Giovanni Resta's site Numbers Aplenty, and noticed that significantly more than 1/37 of all palindromes were divisible by 37. I knew that this was because 37*3 = 111, but to my surprise I couldn't find a single palindrome divisible by 37 but not 111 for a long time (well, a few hours or days). The first one I found was 808808838808808, which quickly led to 191191161191191. Eventually I convinced myself that 5009009009005 was the smallest, but I don't remember how rigorous my proof was. At some time in the future I will do an exhaustive computer search, unless someone else does it first, and post the result.
List of uniform honeycombs
This list is not only outdated and errored, as explained in the heading, but it contains very specific unexplained jargon. The post was written to parallel the research Jonathan Bowers was doing on higher-dimensional uniform polytopes and apply the same methods to search for 3D tesselations. The mysterious names given, such as "chon", "octet", and "squat" are nicknames invented by Jonathan Bowers to describe polytopes and tessellations, formed by contracting the full name down to its initials and adding vowels. These names are commonly called Bowers Style Acronyms and Dr. Richard Klitzing's site here gives a long list of these acronyms with their meanings.
The few exceptions are nicknames invented by me when I stumbled across configurations that were, to my knowledge, new. These nicknames are Goccoticpidsith for "Great cubicuboctahedral tomocubic prismatic disquare tiling honeycomb," Gotactictosquath for "Great tetrahedral cubic tomocubic tomosquare tiling honeycomb," and Wavaccocotsoth (I don't remember what that stood for because I already replaced the name with Wavicoca in one place). All these shapes have now been given better names by Jonathan Bowers, and the list updates are available on this hi.gher.space forum thread (my username is polychoronlover).
Cheryl's Birthday and my solution
Self-explanatory. I thought of the solution in my head and was surprised that everyone else thought it was so hard. Obviously I overestimated the average person's intelligence.
The picture of the problem on paper is from the Guardian article, which in turn got it from Kenneth Kong on Facebook. I don't know if the picture is copyrighted. (I will gladly remove it if I learn that it is.)
How to calculate exact trig values, a post about radical expressions for roots of unity with an unsuspecting title, and its sequels [1], [2]
Ooh, here goes. The first in a series of posts where I did what is in my opinion my most impressive and mysteriously learned work. If only I had given them titles describing what I was actually doing.
I have made several more interesting findings in regards to this topic since making this post, described at the bottom of this section. So, there could be no better time to explain exactly where I learned such a specific way to calculate roots of unity in the first place.
In 2012, I was well aware from my math textbook that cos(45°) = sqrt(2)/2, cos(60°) = 1/2, and cos(30°) = sqrt(3)/2. I had also recently learned that cos(36°) = (1 + sqrt(5))/4 and cos(72°) = (-1 + sqrt(5))/4 from a statement on page 146 of the Penguin Dictionary of Curious and Interesting numbers by David Wells. (The book said that -2 cos(666°) was a good approximation for the golden ratio, but I verified the equation to be accurate to 14 decimals with my calculator. I was starting to suspect the values were actually equal. Later, I proved their equality myself by measuring various lengths in a pentagram.) When I learned the identities cos(a + b) = cos(a)*cos(b) - sin(a)*sin(b) and sin(a + b) = sin(a)*cos(b) + cos(a)*sin(b) (part of a normal high school trigonometry curriculum) I was able to find radical expressions for cosines of every multiple of 6°. On some website that I can't remember, possibly this Wikipedia article, I found the identity cos(15°) = (sqrt(6) + sqrt(2))/4, which allowed me to find a radical expression for the cosine of every angle that was a multiple of 3°. Here are pictures of the paper where I recorded the results: i.imgur.com/wjjEIzP.jpg, i.imgur.com/xWNmrla.jpg.
Around the same time, I was also hearing about how Gauss had proved that the cosine of 17ths of a circle had expressions in square roots, and so did the 257ths and 65537ths of a circle. I was also starting to think less of these calculations in terms of cosines and sines and instead in terms of roots of unity (complex numbers of the form cos(2*pi/n) + i*sin(2*pi/n), in radians of course). Near the end of 2012, I heard of a method to calculate 17th roots of unity by starting with the sum of all 16 primitive roots and solving quadratic equations repeatedly to find the sum of fewer and fewer roots. I learned this method from the book The Book of Numbers by John Conway and Richard Guy. Seeing this derivation, I started to wonder if the cosine of every rational fraction of a circle was expressible using radicals.
I also heard (from the same book? Or maybe Wikipedia?) that the 7th and 9th roots of unity required cube roots when expressed in radicals, and that cos(2*pi/7), cos(4*pi/7), and cos(6*pi/7) were all roots of the same cubic equation with rational coefficients. Using a desk calculator, I found the three elementary symmetric polynomials of these three values, from which I found the coefficients of the cubic equation. Then I used the cubic formula to solve the equation. The resulting radical expression for cos(2*pi/7) matched the expression given on this Wikipedia page and indeed the expression now in my blog post, even though that expression was calculated using a different and more extensible method. I found an expression for cos(2*pi/9) the same way, and even found an expression for 2*cos(2*pi/13) (as x^1 + x^12, where x is a 13th root of unity) by solving a quadratic equation for the sum of six 13th root of unity and solving a cubic equation to break the sum into three pieces. I noticed the similarity between this method and Gauss's method to calculate 17th roots of unity. In general, it seemed that to calculate the pth roots of unity, where p was prime, I needed to start with the sum of p - 1 pth roots of unity (which was always -1). Then I needed to repeatedly break the sum into q smaller sums, where q was a prime factor of p - 1, and find their values by solving a q-th degree polynomial. The coefficients, symmetric polynomials of the sums of roots of unity, could always be expressed in terms of previous sums whose values were already determined.
This method did not suffice to calculate the 11th roots of unity. I would have to solve a quintic equation, because 5 is a factor of 10, but I knew that there was no general quintic formula by the Abel-Ruffini theorem. I knew there was a radical expression for the 11th roots of unity because by then I had read somewhere that roots of unity could always be expressed in radicals. After thinking for a few weeks or months, in early 2013 I came up with a new way, based on the discrete Fourier transform of the roots. This method gave me a new way to find the 7th and 13th roots, and also allowed me to find the 11th roots by avoiding the problem of having to solve a quintic equation. This is the method I describe in my blog posts.
In January 2013, before I knew how to compute expressions for the 11th roots of unity, I found an expression for cos(2*pi/11) on a website called Literka. I was unable to compute the values myself for a long time even after I learned the method described in my posts, because the calculations were too intense (they involved taking a ten-term expression to the fifth power), although I made several close calls. In 2017 I learned enough shortcuts and perserverance skills to compute the values and promptly wrote a post about it, and I am happy to see that the values I found, with the -979, 275, and 220, match the values given on Literka.
But now all these advancements in my knowledge have been outdated by my recent research on the matter. Now I have:
A Python program that finds root-of-unity expressions using the same method that I used,
A "refinement" of this method that gave shorter expressions for the 11th roots of unity,
Another Python program that finds expressions using the refined method,
Lots more root-of-unity expressions, some very long, that are numerically verified to be accurate,
A formula to approximate the average "length" of a root-of-unity expression before I calculate it.
The Python programs can, in theory and assuming arbitrary precision floating point values, calculate the radical expression for the p-th roots of unity where p is any prime. In reality, they tend to falter for most cases past 60th roots; the expressions become impressively long. The lengths vary a lot between consecutive primes, sometimes by orders of magnitude.
I can't wait to make another post presenting all these things!
The bases described in this post are base 37, centered nonary, double-digit base 120, base ϕ, definite length base 10, and a "base" based on the prime factors of a number.
Base 37 is a base that I haven't seen anyone else but me take an interest in. It is quite an arbitrarily chosen base, and I chose it just because 37 is my favorite number.
I think I got the idea for centered nonary from a post about centered ternary on the XKCD forums (or was it the hi.gher.space forums?). I probably chose base 9 just because 9 was close to 10. Unfortunately, I can't find the post.
I first heard of base 120 from this page of the metrologist and polytopist Wendy Krieger, who uses it extensively on her website. I discovered her website from my interest in higher-dimensional geometry. She calls the base "twelfty." Much of what I posted about base 120 came from her website, with some notation changed, most notably to use commas to group digits into pairs of super-digits.
Base ϕ is rather well-known. It is mentioned on Wikipedia, Ron Knott's site, and this hi.gher.space post, for example. I first heard of it when I was playing with the cellular automata simulator Golly written by Andrew Trevorrow and Tomas Rokicki. There was a pattern called "alien counter" and the description said that Adam P. Goucher (also known as Calcyman, the creator of apgsearch and Catagolue) thought the simulation was counting in base ϕ but no one was really sure. I didn't actually learn how base ϕ worked for several more years.
I haven't heard of definite-length bases anywhere else, although I'm sure someone else has come up with them. I first thought of the idea when I was young and wanted to use base 26 to assign each word in English a unique number. I numbered A-Z as 1-26, then realized that base 26 would require them to be 0-25. But to my surprise, my numbering scheme still assigned each string of letters to a unique positive integer. Definite-length base n, made from powers of n and the digits 1-n rather than 0-(n - 1), was born.
Even though I independently discovered the prime-factor "base" representation of numbers, the same system appears in OEIS entry A054841 and was called "Exponential Prime Power Representation" by Walter Nissen. The idea of representing a number in terms of its prime factors is very old. In fact, Gödel encoding is a way to map strings of characters to integers in much the same way as exponential prime power representation. Each character position gets assigned a unique prime number and each character gets assigned a unique exponent of that number. Unlike definite-length bases described above, Gödel encoding works for a language with an infinite set of characters.
The Starter Constant Mystery saga: [0], [1], [2], [3]
These are the posts that I feel the most personal connection to. They involve my findings about the following triangle of numbers:
1
0 1
0 1 2
0 1 6 6
0 1 14 36 24
0 1 30 150 240 120
...
These numbers are generated from the nth powers x^n in a way that I now know is called the inverse binomial transform. My posts call them the starter constants. The OEIS calls them numbers of the form k!*Stirling2(n, k).
These numbers have been studied before. They appear in many entries on the OEIS, under many different descriptions: [0], [1], [2], [3], [4]
In my posts I prove two main things about these numbers: that they can be generated by an inductive formula similar to Pascal's triangle (idea 1), and that they relate to the number of strings of given length using every type of character from a given set (idea 2). All the proofs are my own, and I felt greatly accomplished when I made them. The same proofs may have been found by other people, but they are the pride of my mathematical life right now. The major things I found from outside sources are attributed in the posts. Notably, idea 2 described above came from the description of OEIS post [4]. I was amazed when I read that description, when I had only known the starter constants as relating to powers. Rather than looking at the post to see a proof, I felt challenged to prove the connection myself, which I did and detailed in my first post.
When does n have the same digits as 2n?
In this post I describe an algorithm for finding numbers n that are anagrams of their doubles, 2*n, in base 10. The algorithm was developed by me around May 2015. I was inspired to do it after finding, and skimming, but not reading, this post: https://blog.plover.com/math/dd.html, which I found from the OEIS entry for the sequence: A023086. In fact, my own technique of finding cycles of digits is very similar to what the post describes, and the post goes into more detail on the constraints of the digits.
As you probably can tell, I had a tendency when I was younger to find interesting mathematical facts online, attempt to prove them myself rather than looking at someone else's proof, and post them to my blog without attribution if successful.
Unnamed post about the coordination numbers of hyperbolic tilings and its sequel
In these posts, I discuss regular hyperbolic order-3 p-gonal tilings and the sequences arising when counting the number of polygons a given number of steps from a center polygon.
This sequence is called the tiling's coordination sequence, even though I didn't use that term until nearly the end of the second post. I was inspired to study coordination sequences after seeing a post on Adam P. Goucher's blog that said that the coordination sequence for the heptagonal tiling was 7 times alternate Fibonacci numbers. As usual, I skimmed Goucher's post without trying to find a proof of this phenomenon and proved it myself in my post.
There is also the issue of how I created the images for my posts. The images were created using Paint.NET and the font used for the layer numbers was Frente H1, which I downloaded from a site called Font Squirrel. If I remember correctly, Times New Roman was used for the colored numbers in the first post and Arial was used for the colored numbers in the second post.
The post contains two types of images of tilings. One type gives the entire tiling in red and the other gives each layer in a successive rainbow color. For the first type, I used these public domain images [1] [2] by Tom Ruen and Anton Sherwood, which were widely used on Wikipedia until they were replaced by SVGs. The second type of image was created by hand, although I sometimes used the two images given above to trace the edges on them. It also deserves mentioning that the rainbow palette that I used to color successive layers in my images was inspired by the image in Goucher's post.
Here is a link to the OEIS entry for the coordination sequence of the order-3 heptagonal tiling (listed by them as the coordination sequence for the vertices of the dual tiling): A001354.
Last digits of powers and its sequel
In these posts, I prove various theorems about patterns in the last digits of the decimal expansions of perfect powers. As far as I can remember, I thought up the proofs myself, using nothing more than algebraic manipulation and Fermat's little theorem. Although I played with powers a lot as a kid, all of the results I derive are well-known and I only use basic methods from number theory to prove them. For example, see this brilliant.org post about last digits of powers.
The first post is about how x^n = x^(n + 4) (mod 10) when n ≥ 1 and the second is about how x^n = x^(n + 20) (mod 100) when n ≥ 2. I was going to make more posts about the equivalences of powers mod 1000, higher powers of 10, and eventually mod other bases, but I lost interest.
The "wonderful" theorem I mention in the second post says that if numbers a and b are coprime, then pairs of values (x mod a, x mod b) will only be the same for two different values of x when those two values are separated by a multiple of a*b. This is equivalent to saying that when a and b are coprime, a*b = LCM(a,b). This is simply another basic result of number theory.
This post was an April Fool's joke announcing a "mathematical breakthrough": the discovery of irrational numbers that are prime! Obviously, this is absurd as all prime numbers are integers by definition. At the same time, I also intended to present an actual mathematical curiosity I had rediscovered: a set of real numbers containing all primes, no other integers, and infinitely many (probably irrational) non-integers.
The idea is that Wilson's theorem, a common test for prime numbers named after John Wilson, says that an integer x is prime if and only if ((x - 1)! + 1)/x is an integer. Using the Gamma function, factorials can be defined for every positive real number, allowing for integer values of ((x - 1)! + 1)/x to exist for non-integer values of x. For the sake of a joke I called these numbers "irrational" despite not knowing that for sure.
In fact, the curiosity of non-integer solutions of Wilson's congruence has been noticed by several people before, such as in this college math test from 1999.
Lychrel numbers and its sequel
In these posts, I discuss Lychrel numbers, numbers that never reach a palindrome when iterating the process of reversing the digits and adding to the original number: why we think they exist, why we haven't proved they exist in base 10 although we have proved they exist in base 2, and a heuristic argument suggesting there are infinitely many Lychrel numbers in every base. The first post is the first one that I can remember where I only talk about things other people have discovered instead of presenting my own findings, even if they are rediscoveries or trivial corollaries of others' discoveries. The binary Lychrel number mentioned in the second post I found on Wikipedia. The only insights that were my own were the heuristic argument and the connection to cellular automata.
Unnamed post with a proof about geometry
In this post, I prove that removing four non-adjacent vertices of a cube results in the vertices of a regular tetrahedron. This fact may be useful in chemistry as an aid to drawing molecules with tetrahedral structure, such as methane. In fact, as I mention in the post, I got the idea after watching a chemistry video on Khan Academy with a proof of the same fact. I think the video was this one: https://www.khanacademy.org/science/organic-chemistry/gen-chem-review/hybrid-orbitals-jay/v/tetrahedral-bond-angle-proof.
Even though I was quite proud of it at the time, I now think that my proof was trivial and was presented in an overly long fashion. I don't actually know why I needed to mention the coordinates of a cube (let alone one rotated 45 degrees from the standard coordinate-parallel orientation!). I should have just said that choosing alternate vertices produced four vertices such that every pair lay on opposite ends of a square, and thus the distances between all pairs of vertices was equal, and it followed that the vertices produced four equilateral triangles which were the faces of a regular tetrahedron.
Jonathan Bowers' birthday, plus a program
This post discusses more of the work of Jonathan Bowers, the same guy whose shape names I used back in my "List of Uniform Honeycombs" post. In particular, it talks about Bowers Exploding Array Function (BEAF for short, also called Array Notation), a notation that represents enormous numbers as multidimensional arrays (and beyond) of positive integers. This post may be the most interesting to readers of the blog because BEAF has gained a cult following around the Internet, and is an inspiration for many modern large number notations found on the Googology Wiki, such as Bird's array notation, Hyp Cos's Strong array notation, and Sbiis Saibian's Hyper-Extended Cascading-E Notation.
In my post, I present a Python program that I wrote to "evaluate" Bowers arrays into large numbers, one step at a time. In reality, the program usually either ends up repeatedly decrementing a number larger than one million, but still small enough to be expressible directly as a binary integer inside the computer's memory, until the maximum recursion depth is reached, or otherwise ends up crashing trying to evaluate an expression with such a number in the exponent. This was deliberate; as the post states, the results of evaluating most Bowers arrays are numbers too large to fit in the universe! Even an array as small as {3, 3, 3} evaluates to a number equal to 3^3^3...^3^3, with 7,625,597,484,987 threes. This number couldn't fit in the observable universe if written out in full, and its number of digits couldn't fit, and the number of digits in its number of digits couldn't fit... in fact, you could take the number of digits over seven trillion times and the resulting number would still be too large to write out in full! I mostly wrote the program to showcase the fact that BEAF is a recursive function, and thus computable. Even though BEAF produces numbers so large they have no uses outside of pure mathematics, the arrays can be evaluated step-by-step with a computer. In fact, a computer with infinite memory and time could evaluate any Bowers array all the way down to the final mind-boggling result!
Unfortunately, I abandoned work on the program soon and never got beyond implementing the rules for 2-dimensional arrays. However, I may update it when Bowers' 50th birthday arrives later this November.
I first learned of Bowers Exploding Array Function in 2011 after discovering Bowers' work with polytopes. I learned about linear arrays through Bowers's own web page describing his notation, and I learned about multidimensional arrays from a post on qntm.org and its sequel on everything2.com. But Bowers' notation goes way beyond multidimensional arrays, into super-dimensional arrays (don't ask) and tetrational arrays. I slowly learned how these worked through immersion and from Googology Wiki; it became easier after I learned that each level of tetrational space corresponds to an ordinal number below ε0. Bowers' notation goes even further, into pentational arrays, hexational arrays and beyond. In 2014, I learned that no one understood how these worked, besides possibly Bowers; his webpage never accurately defined them. However, much work has been done on the Googology Wiki about proposing various formal definitions for the various array structures beyond tetrational arrays, a topic that relates to the wild world of ordinals beyond ε0. I will probably discuss these formalizations in a future post some day. In fact, earlier in 2019 Bowers announced that he was planning to make a video series about higher level array structures, so we can still look forward to an official definition of BEAF.
Unnamed post about 2D lattices of points
This post is about 2D lattices of points generated by a pair of basis vectors, where the basis vectors have integer coordinates. In it, I proved that the fraction of all points in Z^2 that are in the lattice is 1/|D|, where D is the determinant of the matrix whose columns are the basis vectors. In the proof, I use the well-known fact from vector calculus that the area of a parallelogram in 3-space is equal to the absolute value of the cross product of its edge vectors (although I mistakenly leave out the absolute value sign). I also write an overly long alternate proof that every point in Z^2 is in the lattice if and only if |D| = 1. Interestingly, I had never taken linear algebra when I made this post (I only knew about determinants and matrix inverses through Khan Academy videos), and I ended up rediscovering by myself the fact that the determinant is a scaling factor of the transformation.
In this post, I use an image that I created with the Khan Academy Processing.js programming tool: https://www.khanacademy.org/computer-programming/new/pjs. The image of the matrix equation was made using Microsoft's Math Input Panel for Windows 8 (however, most equations in later posts were made using latex2png.com).
Higher-dimensional Pascal's Triangle and its sequel
In these posts, I consider how the trinomial coefficients, quadrinomial coefficients, and beyond can be arranged into higher-dimensional generalizations of Pascal's triangle, which are called Pascal's simplices (plural of simplex). I investigate several properties of the simplices, including:
How the numbers can be derived from counting combinations of objects into multiple groups
How each layer of a simplex can be obtained by adding numbers in the layer above, similar to the rows of Pascal's triangle
How the entries on each row of the n-dimensional simplex sum to a power of n
How each entry in the (n + 1)st row of a simplex counts the number of vertices of a uniform n-polytope with simplectic symmetry
How the entries on the interior of each layer sum to one of the starter constants!
I rediscovered all of these properties by myself because I didn't want to be disappointed if I learned that Pascal's simplices had already been investigated. As it turns out, this was in fact the case. Almost every property I mention is described in two Wikipedia articles, Pascal's pyramid and Pascal's simplex. The article "Pascal's pyramid" has references to the object going back to the 1970s. The earliest known publication of Pascal's pyramid seems to be this paper by John and Larry Staib, who appear to be father and son. Even they believe, like I did, that the triangle is so elegant and easy to construct that it must have been discovered before.
The posts mention higher-dimensional simplices, hypercubic tessellations, simplectic tessellations, and partition numbers. I learned about all these topics on Wikipedia (except for partition numbers, which I learned about in The Book of Numbers by John Conway and Richard Guy, rather than being rediscovered by me. The "advanced polytope terminology" in the second post, such as "Biexipetirhombated dodecadakon," is Jonathan Bowers' nomenclature for kaleidoscopical convex uniform polytopes, described on these pages of Bowers and Richard Klitzing: [0], [1]. The textual representation of a Coxeter-Dynkin diagram (o3o3x3o3x3o3o3o3x3x3o) was also developed by Klitzing.
I created some of the illustrations in my post using the Khan Academy Processing.js programming tool and used paint.net to screenshotted and cropped (and in one case modify) them. The code to generate these illustrations is available here: [0], [1], [2]. The other images were created by cropping screenshots of Notepad.
Cyclotomic polynomials and repeating decimals
This post came out of a lifelong interest in using repeating decimals to represent rational numbers. I was interested in this subject ever since my dad showed me how the decimal expansions of different multiples of 1/7 consisted of the same cyclic permutation of 6 digits.
In the post, I mention that in order for an integer p to be the period length of the base-b expansion of 1/a, a must divide b^p - 1. I made this connection myself as a child by studying the repeating decimals of 1/7. I also found out that if a is prime and not a factor of b, the period p must be a factor of a - 1 by considering the possible different cycles that can occur when taking different multiples of 1/a. All of these facts are well-known, and are mentioned on the Wikipedia article for repeating decimals.
I also mention that the primes a with period p are all factors of cb(p), the bth cyclotomic polynomial evaluated at x = p. I learned this after investigating this page from Studio Kamada, which gives the known prime factors of cb(10) for every value of b up to 300000 (!). I discovered the page when studying the factorization of repunits (for which there is a similar page on the same site) and trying to isolate the factors of (10^p - 1)/9 that had period length p rather than some proper divisor of p.
In the post, I mention Artin's Constant and the related conjecture involving full-reptend primes in base b. I learned about these topics from The Book of Numbers by Conway and Guy. The images were created using latex2png.com.