The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].
Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].
Here's something I just learned: the lucky numbers of Euler.
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k² − k + n produces a prime number.
Leonhard Euler published the polynomial k² − k + 41 which produces prime numbers for all integer values of k from 1 to 40.
Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).
The Heegner numbers 7, 11, 19, 43, 67, 163, yield prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.
h/t John Carlos Baez
(@johncarlosbaez) for pointing this out.
🐍 aprxc — A #Python#CLI tool to approximate the number of distinct values in a file/iterable using the (easy to understand) Chakraborty/Vinodchandran/Meel #algorithm¹.
watching #QI. An infinity of mathematicians go into a bar and place an order. "I'll have a pint, a half pint, a quarter pint, an eighth, an ounce.." The barman says "I'll stop you there," and pours out two pints. "The problem with you mathematicians is you don't know your limits" #math#joke
Can someone help me with this #Geometry volume problem?
I'm trying to calculate the volume of water in my storage pond. It is a section of a cone with these dimensions: the bottom is roughly circular and 40 feet diameter; the top is roughly circular and 65 feet diameter. It is 16 feet deep when full.
I probably could've done this in high school geometry class, but that was…er...a very long time ago. #math#FediHelp
Im currently working on trajectory systems and forgot what tan() does and I kinda am embarrassed about that. Also I'm awful at trigonometry apparently. Haven't thought about it in a couple decades!
First authorb paper out in the wild. It's challenging as an independent researcher, but it can be done. This has been a long time coming. Maybe more in the future https://zenodo.org/records/11214976 #paper#math#matheducation#proofs
Should show up in a couple other locations as well hopefully (pending reviews)
"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater ; not because the pleasure it gives (although very pure) is comparable [...] to that of music [...]" – Bertrand Russell (1872–1970) #quote#mathematics#art#maths#math
(1/2) Hands-On Mathematical Optimization with Python 🚀
The Hands-On Mathematical Optimization with Python book by Krzysztof Postek, Alessandro Zocca, Joaquim Gromicho, and Jeffrey Kantor provides the foundation for mathematical optimization. As the name implies, the book is hands-on with Python examples, mainly using Pyomo.
Hi all, I've started the arxiv submission process of my first author paper in the general math category, but it needs an endorser. Apparently the endorser must be someone who has published 2 papers earlier than 2 months ago and less than 5 years ago, in the general math category. Please let me know if you can. Thanks in advance! #math#arxiv#papers
I have a question about the aperiodic spectre tile (or the hat/turtle).
I know that the proof of aperiodicity works by showing that the tiles must fit together in a hierarchical structure that eventually repeats itself at a larger scale. But the larger units aren't literally scaled copies of the spectre. I also know that there is some freedom as to how you draw the edges of the spectre.
Is there a way you can draw the edges that allows you to literally use spectres to cover a larger copy of themselves? If so, is this way of doing it unique?