Tag: math
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213
Since August 1 is the 213th day of the year (2023 not being a leap year), this seems like a good time to share an interesting fact about the number 213. It involves semiprimes, so let’s review what a semiprime is: a product of two distinct prime numbers. 213 is a semiprime because it is…
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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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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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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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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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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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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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Prime signatures
Prime signatures are a way of describing the prime factorization of a number. The prime signature lists the exponents of the prime factors, typically sorted from smallest exponent to greatest (or vice versa). For example, the number 28’s prime factorization is 22 × 7, which can be abstracted to p2q (where p and q are distinct primes) or p12p2 (where p1 and p2 are distinct primes.) The prime…
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Iterative mapping of prime signatures
Perhaps the most elegant way of arranging all of the positive integers in a two-dimensional grid is the pairing function. The grid is populated by placing a 1 in its starting corner, then filling the grid with diagonal stripes: 2 and 3 in the first stripe, 4 through 6 in the second, and so on. Below…
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Prime signature table
A very large table of the first few numbers for each of the first 160 prime signatures.
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