Category: math
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Non-square-free numbers
A square-free number is one that does not contain a square as a factor. One interesting sequence involves non-square-free numbers: that is, numbers that are divisible by a square. The first number in this sequence is 4, which is the smallest integer containing a non-trivial (i.e. not 0 or 1) square. The second number is 8, which is the first…
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Chatting with ChatGPT
In November 2022, a chatbot named ChatGPT went live. A great deal of media coverage followed, reporting on both the bot’s impressive capabilities and its fairly significant downsides. I decided to try it out recently. It is an impressive technological achievement. It is also fairly terrifying. Its ability to understand the prompts you provide and…
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Math & numbers
I enjoy recreational mathematics, numbers, and logic puzzles. On this page you can find some of the mathematical curiosities I’ve found interesting.
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Favorite mathematical sequences
Non-square-free numbers A square-free number is one that does not contain a square as a factor. One interesting sequence involves non-square-free numbers: that is, numbers that are divisible by a square. As of 2023 the highest number of consecutive non-square-free numbers is 18 according to the OEIS. Untouchable Aliquot Numbers Take any integer and add its proper divisors…
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Fibonacci numbers and Pisano periods
The Fibonacci numbers (sequence A000045) form one of the most important and well-known sequences in mathematics. The sequence is very easy to construct: start with 0 and 1 as the first two numbers in the sequence, and after that, every new term in the sequence is the sum of the previous two: 0, 1, 1, 2, 3, 5,…
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Online Encyclopedia of Integer Sequences entries
Relating to Metaprimes A079708: Metaprime binary to standard binary conversion series (first term = 0). A133487: Metaprime binary to standard binary conversion series (first term = 256). Fibonacci numbers and Pisano periods A179390: Modulus for Fibonacci-type sequence described by A015134. A179391: First term in Fibonacci-type sequence described by A015134. A179392: Second term in Fibonacci-type sequence described by…
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Collatz tree / 3x+1 conjecture
The Collatz conjecture (also known as the “3x + 1” or “3n + 1” problem) involves a simple repeated formula, but yields some interesting sequences. In short, begin with a positive integer; if it’s even, divide it by two; if it’s odd, multiply it by three and add one. Repeat this sequence until the result is 1.…
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Why does 0.999… equal 1?
One mathematical question that sometimes causes some confusion is this: “Are 0.999… and 1 the same thing?” The answer is yes, assuming by “0.999…” we mean a zero, then a decimal point, then nines that repeat forever. There are pages and pages of lengthy mathematical proofs available that show why this is so, but my favorite way of showing why 0.999……
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Metaprimes
One day a few years ago, while waiting for my dinner, I started doodling with prime numbers, and I came across a question I found interesting. Here’s how it came about. When translating a number from binary to decimal notation by hand, you add together a series of products. For example: 1110 (binary) = (16…
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