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Furstenberg's Infinitude of Primes; 20 may 2026

Differential Equations of Type Constructors; 6 dec 2025

Multilinear maps don't preserve 0; 1 dec 2025

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dajo42

5001 being divisible by 3 doesnt feel right

dajo42

shinyhappygoth

Shortcuts to determine if an integer is divisible by:

This is a given.

If the last digit is divisible by 2 (a.k.a. even), then so is the whole number.

If the sum of the digits is divisible by 3, then so is the whole number. The recursivity of this means that if the sum has multiple digits, you can add them up again until you get a single digit and see if it's 3, 6, or 9.

Like the rule for 2, but check if the last two digits are divisible by 4.

If it ends in 5 or 0.

If the rules for both 2 and 3 apply.

No shortcut. Alas.

Like the rules for 2 and 4, but check if the last three digits are divisible by 8. (Yes, this pattern keeps going for 16, 32, etc.)

Like the rule for 3, but the sum of the digits (or the sum of the sum of the digits, etc.) must be 9.

If it ends in 0.

noelements-setempty

11. Put alternating + and - signs in front of each digit such that the last digit has a + sign (so for 5001 it's - 5 + 0 - 0 + 1). If the answer is divisible by 11, so is the original number. This is recursive like the 3 and 9 rules.

unnatural-transformations

7: Subtract twice the last digit of your number from the number you get by ignoring the last digit. If the result of that operation is divisible by 7, so is your original number (eg, 91 -> 9 - 2×1 = 7, so 91 is a multiple of 7). As with the rules for 3, 9 and 11 this rule is recursive (2261 -> 226 - 2×1 = 224; 224 -> 22 - 2×4 = 14; 14 -> 1 - 2×4 = -7 and so 2261 is divisible by 7).

It's not a fast way of checking if a number is a multiple of 7 (you honestly may as well just do the division), but it does work.

unnatural-transformations

In fact, I think, something along these lines works for any prime (excluding 2 or 5).

Suppose that p is such a prime. Then, by Bezout's lemma, there exist integers X, Y such that 10X + pY = 1 (and clearly any such X is not a multiple of p). Let N := 10a + b. Then p | N if and only if p | NX. But NX = 10aX + bX = (1-pY)a + bX = (a+bX) + p(aY). So p | N if and only if p | (a+bX).

For p=7 we have, for example, (X,Y) = (-2,3). Hence 7 divides 10a+b if and only if 7 divides a-2b. For p=3 we have (X,Y) = (1,-3). So 3 divides 10a+b if and only if p divides a+b (which implies the standard sum of digits test given above). For p=11 we have (X,Y) = (-1,1). So 11 divides 10a+b if and only if 11 divides a-b (which implies the alternating sum of digits test given above). And once you find a suitable (X,Y) you can derive divisibility tests for larger primes as well.

#math #sick!

transgenderer

Lots of people find math tedious so they think of math people as unusually willing to tolerate tedium. But math people don't find math tedious; math people are unusually UNWILLING to tolerate tedium, and only want to focus on the "interesting part", the underlying structure. to be clear, this is a vice. lots of good things lie behind tolerating tedium

#math #possibly the realest shit she's ever said

bubbloquacious

A category does not have a set (or class, whatever) of objects, it has a groupoid of objects, and a set of isomorphism classes.

bubbloquacious

Talking to @shocked--kwarrtz about this actually unconvinced me of this post. So the reason I said this is that the functor that takes a category to its 'underlying set' of objects does not preserve equivalence. That is, an equivalence of categories need not induce a bijection on objects. More elegantly, the underlying set functor is not a 2-functor, so its domain can't really be said to be the 2-category Cat of categories, but rather the 1-category StrCat of strict categories.

That being said, although the functor that maps a category onto its groupoid core does preserve equivalence, it is (counter to what I thought making the op) still not a 2-functor unless you view Cat as a (2,1)-category (that is, you restrict to only the invertible natural transformations).

This is all inspired by the statement you'll see studying ∞-categories that an ∞-cat does not have a set of objects, but rather an ∞-groupoid (/anima/space/homotopy type) of objects. However, I think this only holds up if you only consider natural isomorphisms instead of more general transformations (as many introductions to ∞-cats do), as otherwise mapping an ∞-cat onto its groupoid core is again not actually the right kind of functor of (∞,2)-categories.

#math #infinite cats

A category does not have a set (or class, whatever) of objects, it has a groupoid of objects, and a set of isomorphism classes.

#math #infinite cats

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Are all conditionally convergent series that don't converge absolutely 'equally' conditionally convergent? Or is there some way to measure whether one series converges more conditionally than another?

#math #have had this question for a good long while

Got bangs today and my forehead is so warm

perdvivly

Clifford the big red algebra

#math #yep

perdvivly

Learning things is kind of like an art project where the artefact you produce at the end of it is a mind that knows things

bubbloquacious

Homotopy Types: a type should really be an isomorphism class, not an object in it

∞-Groupoids: too much focus on the algebraic structure

Kan Complexes: too model-specific

Spaces: they contain so much less inherent information than what we usually think of as spaces

Anima: you just really cannot trust the international mathematical community not to pronounce this 'enema'

??? what's the solution here folks

locally-normal

We have a planet called Uranus, we just have to have the maturity to accept potential mispronunciations. Thus, anima. Though I like to say space despite everything.

m---a---x

I think except for spaces all of them are ok. Regarding types, it can make sense to talk about specific computation of types

neatlymanifolded

What do we mean by "spaces" here? from the intro sentence i assume not just topological spaces.

locally-normal

Spaces meaning CW complexes up to homotopy, equivalently CGWH spaces up to homotopy, equivalently all those other things. Whether these should be referred to as spaces is contentious in the homotopy theory community.

neatlymanifolded

How is this different to a homotopy type? At least within geometric topology, "homotopy type" and "CW complex up to homotopy" are synonyms, unless you suddenly care about non-CW spaces.

bubbloquacious

I think referring to them as 'homotopy types' is like referring to sets as 'cardinal numbers'. It's the class, not the representatives!

#math

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government-cy-op

Just found 2 copies of a 19th century naturalism magazine and LOOK AT THEIR LITTLE ICON

I think that all the data of a theory of scissors congruence might be able to be captured in a single morphism of topoi... Stay tuned I guess

#math #scissors congruence

Really fun how you can read it both as "homotopy (type theory)" and "(homotopy type) theory"

#math #infinite cats

Homotopy Types: a type should really be an isomorphism class, not an object in it

∞-Groupoids: too much focus on the algebraic structure

Kan Complexes: too model-specific

Spaces: they contain so much less inherent information than what we usually think of as spaces

Anima: you just really cannot trust the international mathematical community not to pronounce this 'enema'

??? what's the solution here folks

#math #infinite cats #i'm thinking of calling them 'souls' if i ever do a homotopy theory write up #it's the english translation of 'anima' plus it starts with S #and it gives the idea of being a sort of distilled form of a space

bubbloquacious

The rational numbers are such a fake topological space

bubbloquacious

The fact I read that prompted this thought btw is that Q is homeomorphic to its own iterated product as a topological space, but not as a locale. Freaky stuff!

#math #still working on the intuition for this #it's to do with the local compactness i think

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bubbloquacious

In case you're not convinced enough that non-commutative algebra is super weird and fucked up: For a commutative monoid, being cancellative is sufficient to conclude that it is embeddable in a group. For a non-commutative monoid this is not the case!

michaelrotonal

fr? name an example

bubbloquacious

It's pretty hard to find them! Malcev proved this result in 1937 by taking the free monoid on generators a, b, c, d, x, y, u, v, and quotienting by the relations ax = by, cx = dy, au = bv.

#math

Anyone have any thoughts on the new (g⁺)⁺ video?

#math

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