A curious math problem I came up with: given a target, what's the fewest digits an integer must have (in a given base) to contain all integers from 0 to the target, as substrings?
e.g. for a target of 19 a candidate representative would be 1011213141516171819 in base 10, that has 19 digits. Can it be done in less, or is $\sigma_10(19) = 19$?
Can we find a general rule? Any properties of this function?
After eight years of undergraduate education (it's a long story), I finally graduated more or less by chance with a BA in Philosophy. But my interests were in epistemology, symbolic logic, and the philosophers of the Enlightenment, so I never did take a course specifically about Plato or Aristotle.
I must, therefore, cast my bread upon the waters and trust search engines to eventually find the few people who might be interested lurking in the long tail of the internet.
When Thomas Oliver and Kyu-Hwan Lee used machine learning techniques to predict the ranks of elliptic curves with high accuracy, they noticed hidden oscillations reminiscent of bird murmurations. That pattern was not noticed by mathematicians before, and an explicit formula for those was found by Nina Zubrilina.
2024 is also an abundant number (that is, the sum of its divisors is greater than the number [1]) and a Harshad number (the number is an integer which is divisible by sum of its digits, here 2024 = 8 x 253 [2]).
The sum of the divisors (one, two, three, six, twenty-nine, fifty-eight, eighty-seven, and one hundred seventy-four) of the number of letters in this sentence, which wishes you a very Merry Christmas, is three hundred sixty! #Christmas#wordplay#NumberTheory
Overdue intro post:
• I live in the US Midwest
• Married with cats
• This is my music alt; I make music with #ModularSynths in #eurorack hardware and in software, plus some regular instruments, including guitar & Chapman Stick
• Hit me up for modular synth lessons
• I've made a living in #software for 30 years, including #AI work recently, but am on a hiatus
• I studied #physics, and still take an interest in the frontiers of the field
• I study #NumberTheory for fun
• I am #ActuallyAutistic
Prime Numbers: The Most Mysterious Figures in Math
A fascinating journey into the mind-bending world of prime numbers.
Mathematicians have been asking questions about prime numbers for more than twenty-five centuries, and every answer seems to generate a new rash of questions.
Our new preprint is out on the arxiv! I have really wonderful coauthors and I'm really very proud of this project for many reasons (some of them deeply personal).
I'm probably gonna write a blog post (or thread) about this project soon but in the meantime please check it out!
It turns out that 2 + 2 = 4 isn't quite as simple as we were lead to believe. Having dealt with more #STEM bigots than I'd have preferred, I can vouch for the - ahem - fact that even numbers can be rather subjective.
Oh, and there are some interesting musings on #AI and #MLMs too.
Okay so maybe I'll do another #Introduction since it seems timely. I'm Anton, I'm from the #Philippines and I'm currently a #PhD student in #Mathematics at University of Vermont.
Within mathematics I specialize in #NumberTheory and within number theory I specialize in #ArithmeticGeometry. If I have to be even more particular, I work on automorphic forms, Galois representations (including p-adic Hodge theory), Shimura varieties, and arithmetic manifolds which cannot give rise to Shimura varieties. All of these things which seem rather broad are related to and somewhat unified by what is known as the Langlands program.
I also have a master's degree in #Physics; I currently do not do research related to physics (though some of the math that I use, such as the theta correspondence, were inspired by physics) but I have a soft spot for #MathematicalPhysics (maybe in the future I'll be able to work on that).
Besides math I like a lot of stuff especially #CuteAnimals and especially #Bunnies. I also like #VideoGames especially of the #JRPG genre, though I'm both too busy and too poor currently to play them. Like a lot of Filipinos who grew up in the 90's and 2000's I like #Anime and #Manga (especially from that era). I also like #Art by which I just mean anything that looks cool to me. I mainly want to discuss #Math on here, but by no means will I be posting about only that.
Sidon’s Problem is an interesting problem in number theory (recently solved, see e.g. [1]). It asks how dense a set S of integers can be without containing any solutions to s1 + s2 = s3 + s4 (aside from the trivial solutions {s1, s2} = {s3, s4}), where s1, s2, s3, s4 ∈ S. This and certain generalizations have come to be known as Sidon’s Problem.