For example, the "direct proof" in this paper is six paragraphs long.

> Wow, I was already impressed with the new comment feature on erdosproblems.com and how it's already been used to solve some of the problems. Excited to see if AI can make a meaningful contribution here.

Since then, there has been some discussion of GPTPro finding a bunch of references, thus enabling many of the problem statuses to be changed from "open" to "solved". But it seems that LLMs couldn't find the right reference for this problem.

But there was a different meaningful contribution from AI here instead.

John Horton Conway:

> It's a funny thing that happens with mathematicians. What's the ontology of mathematical things? How do they exist? In what sense do they exist? There's no doubt that they do exist but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, having been a mathematician all my life. How can things be there without actually being there? There's no doubt that 2 is there or 3 or the square root of omega. They're very real things. I still don't know the sense in which mathematical objects exist, but they do. Of course, it's hard to say in what sense a cat is out there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can't push a cat in a direction it doesn't want to go. You can't do it with a number either.

> Given the recent explosion of interest in streaming data and online algorithms, clustering of time series subsequences, extracted via a sliding window, has received much attention. In this work we make a surprising claim. Clustering of time series subsequences is meaningless. More concretely, clusters extracted from these time series are forced to obey a certain constraint that is pathologically unlikely to be satisfied by any dataset, and because of this, the clusters extracted by any clustering algorithm are essentially random. While this constraint can be intuitively demonstrated with a simple illustration and is simple to prove, it has never appeared in the literature. We can justify calling our claim surprising, since it invalidates the contribution of dozens of previously published papers. We will justify our claim with a theorem, illustrative examples, and a comprehensive set of experiments on reimplementations of previous work. Although the primary contribution of our work is to draw attention to the fact that an apparent solution to an important problem is incorrect and should no longer be used, we also introduce a novel method which, based on the concept of time series motifs, is able to meaningfully cluster subsequences on some time series datasets.

Several commenters here seem to ask "okay, so then what's the right way to cluster windows of timeseries??" Perhaps the final sentence of this abstract suggests a solution in that direction?

In case anyone was curious, https://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nomi... suggests that ~17 countries have a GDP of less than $2 billion per year. Seeing as how there are 193+ countries, this means that the LAPD budget exceeds the GDP of fewer than 10% of countries. (The median country GDP is ~$50 billion per year.)

For some extra context: while these 17 countries include some very poor countries, the primary reason that they have such small GDPs is their small population. Their combined population is approximately the same as the city of Los Angeles.

But in any case, Medicare is literally denied to prisoners, which is an example of ... denying welfare benefits for antisocial behavior?

For some context, Wikipedia says:

> In the United States, depending on the context, the term "welfare" ... can also include social insurance programs such as unemployment insurance, Social Security, and Medicare.

https://en.wikipedia.org/wiki/Welfare_spending#United_States

This is not the actual increase of the benefit amount. In particular, it appears the cost of living adjustment this year is 2.5%. I have been unable to find statistics on how many people/households actually receive the maximum amount, but I don't have a particular reason to believe it is large. (The average benefit amounts are significantly below the maxima.)

Tldr: the average SNAP benefit amount received by people has increased and will increase by significantly more than $1/month.

> Burglary is generally considered a property crime, but an array of state and federal laws classify burglary as a violent crime in certain situations, such as when it occurs at night, in a residence, or with a weapon present. So even if the building was unoccupied, someone convicted of burglary could be punished for a violent crime and end up with a long prison sentence and “violent” record.

The article does not state this explicitly, but it suggests that someone who burgles a residence at night with a weapon should not have a long prison sentence, if the residence turned out to be unoccupied. (Perhaps even if it was occupied but the occupants were not "physically harmed"?)

In case anyone was curious, this is a significant understatement. It is currently believed that the Earth had no ice caps for its first 2 billion years, before the Huronian glaciation, and then again no ice caps for another ~1.5 billion years.

https://en.wikipedia.org/wiki/111th_United_States_Congress "gave the Senate Democratic caucus sixty votes, enough to defeat a filibuster in a party-line vote."

https://www.nytimes.com/2009/07/01/us/politics/01minnesota.h... "providing Democrats with something they had long hoped for: 60 votes, and thus at least the symbolic ability to overcome filibusters."

Out of curiosity, I tried to find examples where many people died of a flu outbreak in nursing homes. Here are some articles I found:

This article describes a nursing home in Sheffield where 9 of 60 residents died from the flu during the 1997--1998 season: https://pdfs.semanticscholar.org/e54b/27d527dd0612d0899f22ea...

This article describes a nursing home in Honolulu where 6 of 37 residents died from the flu during the 1989--1990 season: https://www.ncbi.nlm.nih.gov/pubmed/7657975

This article describes a nursing home in Los Angeles where 8 of 101 residents died from RSV in 1979: https://www.ncbi.nlm.nih.gov/pubmed/6527041

This article describes a nursing home in Devon where 4 of 50 residents died from RSV within one week: https://www.ncbi.nlm.nih.gov/pubmed/6736667

In case anyone is curious, this appears to refer to the following event: https://en.wikipedia.org/wiki/Binghamton_shootings

https://lif.blob.core.windows.net/lif/docs/default-source/de... https://www.un.org/development/desa/publications/internation...