Welcome to the mathematics section
of the Wikipedia reference desk.

Select a section:

WP:RD/MAWP:RD/MA

Want a faster answer?

Main page: Help searching Wikipedia

How can I get my question answered?

Select the section of the desk that best fits the general topic of your question (see the navigation column to the right).
Post your question to only one section, providing a short header that gives the topic of your question.
Type '~~~~' (that is, four tilde characters) at the end – this signs and dates your contribution so we know who wrote what and when.
Don't post personal contact information – it will be removed. Any answers will be provided here.
Please be as specific as possible, and include all relevant context – the usefulness of answers may depend on the context.
Note:
- We don't answer (and may remove) questions that require medical diagnosis or legal advice.
- We don't answer requests for opinions, predictions or debate.
- We don't do your homework for you, though we'll help you past the stuck point.
- We don't conduct original research or provide a free source of ideas, but we'll help you find information you need.

Ready? Ask a new question!

How do I answer a question?

Main page: Wikipedia:Reference desk/Guidelines

The best answers address the question directly, and back up facts with wikilinks and links to sources. Do not edit others' comments and do not give any medical or legal advice.

December 31

Lower bound for totient function

Euler's totient function#Growth_rate has a lower bound, involving $e^{\gamma }$ and $\log \log n$ . What is the lower bound if $gcd(n,6)=1$ ? Bubba73 ^{You talkin' to me?} 20:07, 31 December 2025 (UTC)[reply]

Does that make it a factor of 3 larger? Bubba73 ^{You talkin' to me?} 02:13, 1 January 2026 (UTC)[reply]

I assume that you want to prove (or disprove) that

\liminf _{n\to \infty ,\gcd(n,6)=1}{\frac {\varphi (n)}{n}}\log \log n=3e^{-\gamma }.

The proof of the unrestricted version is given in section 22.9 of Hardy & Wright, with Theorem 328 being for the limes inferior of the growth of the totient function while Theorem 323 is for the limes superior of the growth of the divisor function. Their proof is based on the prime factorization of the argument

n,

which offers some hope it may be tweaked to take your restriction into account. ‑‑Lambiam 11:42, 1 January 2026 (UTC)[reply]

Thanks, I have that book and can look it up. Actually I'm trying to see where the upper bound on phi(6n) crosses the lower bound of phi(6n+1). Bubba73 ^{You talkin' to me?} 21:16, 1 January 2026 (UTC)[reply]

Resolved

Bubba73 ^{You talkin' to me?} 02:06, 3 January 2026 (UTC)[reply]

January 3

How has computers increased our knowledge of math?

Was the 1970s the last decade of no-computers? So pencil and paper math hit a peak in the 1970s? So when computers came around, what areas of math increased the most? Any areas had very little changes? The peak of math must have sped up a bit with the invention of computers. Wonder what insight you guys have on that. I do however, suspect not much has changed in math from 1910s to 1970s, or the period with Srinivasan. But maybe each category and subcategories of math has their own increased/decreased pace. ~2025-43645-51 (talk) 03:43, 3 January 2026 (UTC).[reply]

The history of the use of programmable digital computers for scientific computation coincides with the history of these computers, dating to the 1940s. Before that, mechanical calculators were heavily used for scientific computation. But the practice of mathematics, while it informs the procedures used for computation, hardly involves any heavy-duty computations by itself, and the impact on the body of mathematical knowledge has been minimal for most of the 80 years or so since electronic computers became a thing. Even today, almost all cutting-edge mathematical research uses no more than biological brain power, assisted with pencil-and-paper or chalk-and-blackboard scribbling.

Lately, this has started to shift a bit, not through computing with numbers but through computing with mathematical expressions – not so much computer algebra, which is somewhat helpful but hardly advances our knowledge, but through the use of computers to complete or check the validity of very complicated proofs. For a list of results, see Computer-assisted proof § Theorems proved with the help of computer programs and note that the majority were established in this century.

At least one breakthrough result in advanced mathematics has been announced in which an unknown pattern was discovered through the use of AI.^[1] Some believe that AI will soon open a floodgate for important new mathematical results, but for now we don't see more than a trickle.

All this does not mean that the body of mathematical knowledge has not advanced in the last century. On the contrary, there have been spectacular advances, and it feels as if the pace and scope of breakthrough results is accelerating. But this has mainly been achieved without any assistance of computers other than in typesetting the papers publishing the results. ‑‑Lambiam 10:05, 3 January 2026 (UTC)[reply]

But the practice of mathematics, while it informs the procedures used for computation, hardly involves any heavy-duty computations by itself, and the impact on the body of mathematical knowledge has been minimal for most of the 80 years or so since electronic computers became a thing. This is farcically false. ~2025-31850-11 (talk) 12:28, 4 January 2026 (UTC)[reply]

Specifically, the practice of mathematics is and always has been infused with an experimental aspect (this is how people develop intuition, by looking at examples); access to computers has magnified and simplified this to an unimaginable degree. Furthermore, throughout many areas of mathematics, it is commonplace to give case-based proofs that reduce theorems to “just a finite check”; such finite checks can run over millions or more cases. Fifty years ago, the proof of the 4 color theorem was radical and contentious; today, proofs embedding larger computations are commonplace. This doesn’t even get into the vast areas of computational math that only really came into existence since computers became widespread. ~2025-31850-11 (talk) 13:31, 4 January 2026 (UTC)[reply]

I think it is nevertheless true that for, as I wrote, most of the 80 years or so since electronic computers became a thing, the impact has been minimal. And while we know that Gauss for did some heavy calculations and some mathematicians (e.g. Ludolph van Ceulen) spent a large part of their lives calculating ever more digits in the decimal expansion of

\pi ,

the vast majority of mathematical research did not involve heavy-duty calculations. The amount of computation that has gone into the proof of the four-colour theorem is impressive, but in the end the result added not much more than a drop to the pool of mathematical knowledge. ‑‑Lambiam 18:49, 4 January 2026 (UTC)[reply]

Fractal § History points out that computer graphics have popularised, and possibly catalysed, the study of fractals. -- Verbarson ^talk_edits 13:33, 3 January 2026 (UTC)[reply]

Definition of a hemicube

The hemicube (geometry) article states that a hemicube has 3 faces (which are square), 6 edges and 4 vertices, which sides from Mathworld's illustration, though the description ("produced by cutting a cube in half with a plane that passes through 2 opposite corners and the midpoints of 2 edges") matches. Is there a discrepancy in the article? Thanks, cmɢʟee τaʟκ (please add {{ping|cmglee}} to your reply) 19:04, 3 January 2026 (UTC)[reply]

I think that the top "square" of the Wolfram "hemicube" is actually a rhombus and its side length is greater than that of the unit square on the bottom. JRSpriggs (talk) 01:40, 4 January 2026 (UTC)[reply]

Exactly. Therefore, it seems to me that the definition given doesn't fit the description "It has three square faces, six edges, and four vertices. It has an unexpected property that every face is in contact with every other face on two edges, and every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets.", and the infobox. cmɢʟee τaʟκ (please add {{ping|cmglee}} to your reply) 08:53, 4 January 2026 (UTC)[reply]

It appears to me there are two very different mathematical objects being talked about using the same name, and confusion ensues. The object described in the Mathworld article and in Michon page given as a reference is a convex polyhedron which (according to my calculations) can be described as the convex hull of (0, 0, 0), (0, 1, 0), (0, 1, 1), (1, 1, 0), (1, 1, 1), (1, 1/2, 0), (0, 1/2, 1), alternately as the solid bounded by x≥0, z≥0, x≤1, y≤1, z≤1, 2y≥x+z. There are 7 vertices and 6 faces. Four faces are quadrilaterals and two are triangles. Euler's polyhedron formula then gives the number of edges as 11. It appears that much of the Wikipedia article is talking about what you get when you identify opposite points on the surface of a cube, giving a subdivision of the projective plane. This would have 3 faces, 6 edges and 4 vertices. Note that Euler's polyhedron formula no longer holds here because we're not talking about a polyhedron anymore. In any case, I'm not satisfied that notability criteria have been satisfied for either object; as far as I know Mathworld isn't considered a valid source for notability, and the cited reference appears to be just someone's personal web-page. So perhaps the issues can be resolved by submitting the article to AfD. --RDBury (talk) 12:24, 4 January 2026 (UTC)[reply]

Conway et al. give yet another, very different definition:

The n-dimensional hemicube is the convex hull of alternate vertices of the n-dimensional hypercube.^[2]

None of these is what I would have expected, which is that

\{(x,y,x)\in [0,1]^{3}:x+y+z\leq {\tfrac {3}{2}}\}

would be a solid hemicube (one of whose faces is a regular hexagon). ‑‑Lambiam 17:55, 4 January 2026 (UTC)[reply]

I would have expected that too. An explanation is that if you're ordering by simplicity, say by the number of faces, you might name the hexahedron before the heptahedron. RDBury (talk) 23:14, 4 January 2026 (UTC)[reply]

January 5