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Funnily enough, it was almost discovered several years earlier. The science fiction author Arthur C Clarke wrote in “The City and the Stars” a passage that, as an aside, describes a mathematician looking for patterns in the primes by arranging them in a spiral grid. But he never actually tried doing this himself, and so never actually saw the pattern.

The simplest explanation would be the fact that the Riemann zeta function is also equal to the infinite product of 1/(1 - p^{-s}) for all primes p. The proof is rather accessible, see https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_for... . That’s sort of the simplest result that shows a relationship between primes and the zeta function. That’s what this article builds on, but doesn’t give that actual result until about a quarter of the way through.

I skimmed the article, but the next third of the article seems to be devoted to using this relationship between the zeta function and prime numbers to prove the prime number theorem, which is a theorem approximating how many primes are less than or equal to any given number N.

The final third goes into how to get increasingly accurate approximations for the number of primes less than N, ending on the fact that Gausses approximation is in some sense the “best”, but only if the Riemann zeta functions zeroes lie on the critical section.

If you just want a general primer on why the zeta function has anything to do with primes, the product formula might suffice. In which case, the proof on the Wikipedia page might be a better read. The derivation in the article focuses on the general setup that is later built on to prove additional things

Curiously, I relate to both. I have a certain degree of fear or trepidation of sharks when swimming in the ocean. I feel vulnerable, and keep imagining a shark coming up under me or behind me. But I’m also a certified diver and have exactly none of those fears when diving or snorkeling! I think it’s because I have reduced awareness of what’s around me when swimming in the surface. In a sense it’s coming from the same place as a fear of the dark.

Elon Musk is responsible for the deaths of hundreds of thousands of people, thanks to the dismantling of USAID and US foreign aid. Not hypothetical deaths in the future, people who have died in the last year because Musk cut off their supply of medication and nutrition.

Sure, you could argue it was going to be dismantled anyway under this administration. But I think that’s pretty close to the “just following orders” excuse. Which falls especially flat when it was a task he volunteered for!

And I don’t want to understate the harms of other AI CEOs, but in terms of direct, quantifiable deaths, Musk is pretty clearly the most evil.

I don't give much credence to new allegations. Where were these allegations when he was alive

It’s worth noting that it’s common for it to take years or decades for victims to speak out against an abuser. Especially when the victims are children. Especially when the abuser is a prominent figure, like the literal King of Pop.

I’m not going to try and convince you that these allegations are credible (though I believe they are), I just want you to think about how a child victim might behave in that situation. There’s almost never any objective evidence or 3rd party witnesses of abuse. It’s almost always the word of one person against another. And it may be years before a child victim even fully understands what was happening, and years beyond that to come to terms with it.

That article was written 16 years ago and misses a lot as a result. it doesn’t mention the new allegations, including those documented in Leaving Neverland. It also glosses over the accusations in the 1990s. It’s papered over that those allegations were lies, but he settled the lawsuit with them for over $23M.

That’s a very fair point. Frankly I don’t know enough about the Artemis mission and general path, and would like to learn more. I’m certainly open to the argument that NASA’s budget isn’t properly allocated to the right priorities. I was responding just to the classic argument of “why spend money on NASA when we could be spending on …”

This argument comes up a lot, about whether a space program is “worth it” in some sense. One problem I’ve found is that these discussions often treat this in the abstract. And then we get into the nature of human endeavor, the economic benefits of that R&D, etc.

Let’s talk about this in terms of practicalities. The NASA budget for 2026, per Wikipedia, is $24.4B. I often find it hard to really reason about the size of federal budgets, and the impact on tax payers, but I have a thought experiment that I think helps put it into perspective. Suppose we decided to pay for the NASA budget with a new tax, just for funding NASA. And we did that in the simplest (and most unfair) possible way: a flat rate. Every working adult in the US has to pay some fixed monthly rate (so excluding children and retirees). Again, per Wikipedia, that’s around 170M people. Take the NASA budget, divide by 170M, and you get … $11.96/month.

Obviously, there’s lots of flaws in this. That’s not we pay for NASA, we have income tax as a percentage with different tax brackets. But it is a helpful way to frame how much a country is spending, normalized by population. And I think it puts a lot of things in perspective. $11.96/month is comparable to a streaming service. And we talk a lot about whether NASAs budget is better used for other purposes, but we don’t do the same thing for a streaming service.

Hell, look at US consumer spending: https://www.bls.gov/news.release/cesan.nr0.htm (note that that spending is in dollars per “consumer unit,” which is I think is equivalent to an adult US worker, but there might be some caveats). Based on that, the average US consumer spends around $26.17/month on “tobacco products and smoking supplies”. I just feel it’s a little silly to worry about the NASA budget when the US consumer spends twice that on what is objectively a luxury good. At least NASA won’t give you cancer.

Just comes off like a puffed-up pigeon strutting around the stage, trying to cover up his own insecurities. The true great military leaders dont act like this. Even Patton, who has that kind of reputation, was far more thoughtful. On the journey to invade North Africa, Patton took the time to study the Quran, to better understand the people of the land he was to fight over.

I doubt Hegseth even knows what it means to give quarter, probably just said it because he thought it sounded tough

Voter ID is often touted as an important part of election security, but when you look at the threat model of elections it just doesn't do much. Think about how you would try to cheat at an election. The common methods are things like ballot stuffing, throwing out votes, discouraging people from voting, etc. Examples include spreading disinformation about what day voting is happening, seizing ballot boxes and replacing them with forged ballots that favor your candidate, or calling in bomb threats to polling places. These are not prevented by voter ID requirements.

The only thing voter ID prevents is voter impersonation. It prevents you from finding someone else's name and polling place, going there, pretending to be that person, and submitting a vote on their behalf. But that threat doesn't really scale. Even if you assume no one at the polling places notice you coming to vote over and over under different names, a single person could probably only do this a few dozen times on election day. To scale that you would need more people; and every person you add to the scheme increases the odds of someone slipping up or getting caught. But the real issue is if any of the people you are impersonating try to vote! While election officials don't record what people voted for, they do record who voted, and the ballot counting process will automatically note that people voted multiple times. So you would have to figure out some way to gather a database of a large number of people you know aren't going to vote, and get a bunch of people to turn up at a bunch of polling places under those names. It's just not practical to do, when elections are decided by thousands or tens of thousands of votes.

how difficult it supposedly is to have an ID (which is weird when you look at how other countries run elections)

The devil is in the details. I don't trust that the groups drafting Voter ID legislation are doing so in good faith. For example, North Dakota passed a voter ID law years ago. It stated that you needed a valid state-issued ID that included a street address. Sounds fine, right? The problem is that most homes on Native American reservations don't actually have street addresses. Tribal members use P.O. boxes for mail, and that P.O. box is on their driver's licenses. This was brought up when the law was proposed, but it passed anyway. The Spirit Lake Nation and the Standing Rock Sioux tribes had to sue in federal court. They were eventually successful, but it took years, and in the meantime the 2018 midterms were held with many Native Americans literally unable to vote.

See https://www.npr.org/2020/02/14/806083852/north-dakota-and-na...

The entire enterprise of AI for medical advice reminds me a lot of the early 20th century. When X-rays and radioactivity were first discovered, industry rushed to commercialize it. You could get an X-ray in a shoe store to see how your shoe fits! People were putting radium in water and selling it as some sort of curative. Radium was put in paint to make things glow in the dark. Thorium was put into toothpaste. All in this endless rush to commercialize a technology that had captured the public interest without any particular concern for its efficacy.

I'm not saying AI causes cancer, but this rush to sell something in the medical space before proper testing and evaluation really feels similar. And the common refrain I hear is "this so much cheaper than going to a doctor, this will help give access to medicine to those who cannot afford it." Which actually makes it more concerning in my mind. At this point AI is a multi-trillion dollar industry. For-profit companies providing unregulated, under-studied services, targetting people who might not be able to afford standard medical care, doesn't come off as altruistic; it comes off as predatory.

Ummm, is it a good idea to use AI for malware analysis? I know this is just a proof of concept, but if you have actual malware, it doesn’t seem safe to hand that to AI. Given the lengths of anti-debugging that goes in existing malware, making something to prompt inject, or trick AI to execute something, seems easier.

It indeed sucks for the honest workers like your friends who are losing funding because the CIA can't help itself.

So you find an organization filled with aid workers who are dedicating themselves to saving lives, with some instances of CIA infiltration. And the Trump administration, which is fully in charge of both the CIA and USAID, decides the right thing to do is ... get rid of the aid workers?

What do you think is the moral thing to do here?

I'm not saying it's "better" in the moral sense, but from the point of view of the dominant, it's definitely more effective

By what metric does the Belt and Road Initiative provide more soft power than USAID? Do you have any evidence of this?

So, you do the next best thing which is to plant covert agents in NGOs. That's was the real purpose of USAID

That’s offensive to the men and women who worked hard as part of USAID and other foreign aid programs to help others. My wife didn’t spend 2 years in the middle of nowhere in Zambia teaching children to spy on them. My friends didn’t spend 4 years in Mongolia to spy on them.

An American belt and road initiative would be politically impossible.

I think you misunderstand soft power if you think the belt and road initiative is better. The belt and road initiative largely builds infrastructure to aid Chinese interests and locks countries into loans, while providing minimal employment to the locals.

Go to any Sub-Saharan African country, for example, that have benefited from the belt and road initiative and poll them on their opinions of the United States and China. It's not even a competition.

So instead, you have those timid humanitarian aids program which largely served as intelligence and subvertion network.

Those programs have saved millions of lives. Hell, PEPFAR alone (Presidential Emergency Plan for AIDS Relief) is estimated to have saved 25 million lives. Millions of vaccines have been delivered, millions of children provided childhood nutrition.

Another problem is the US is broke.

USAID cost next to nothing compared to everything else in the budget, these arguments about tightening our belt is disingenuous at best. The USAID budget was less than $45B a year. If we paid for that with a flat tax distributed evenly across all US taxpayers (the least fair way to do it!), that would come out to ... $24.50/month/taxpayer.

I genuinely don't get it. I just don't understand how billionaires think.

Everyone knows why he bought the Washington Post: it was for clout and prestige. Just like how the titans of industry built opera houses and libraries in centuries past. You aren't buying it to make a profit. You take care of something valued by society, and you win some respect from society. Conversely, if you burn that thing to the ground, society will hate you.

So why is the profitability of the Washington Post such a concern all of a sudden? Sure, they lost $100M in 2024, but Bezos didn't buy the Post to make money! And it's not like money is tight. Bezos is worth over $250B; in the last few days alone the jump in AMZN stock increased his net worth by over $5B. If he were to hand that $5B over to the Washington Post, they could keep on losing money at that rate for another half of a century! The article makes this exact point in the last few paragraphs.

If Bezos was genuinely concerned about alienating Trump or whatever, why not just sell the Post? Why try to undermine it like this? You are pissing off the people who like the Post, and I don't think the people who hate the Post are really going to care.

I don’t know much about chemistry, but is there a reason why they are using CO2 as the gas medium instead of something else? I was thinking ambient air would be readily available, and you don’t have to worry about suffocating people if it ruptures. Is CO2 particularly efficient to compress?

Frankly, it’s not a lack of arenas that is holding Go back. It’s the fact that, in 2025, we have a language with a runtime that is neither generational nor compacting. I can’t trust the runtime to perform well, especially in memory-conscious, long-running programs.

pretty sure the process I've seen most places is more like: one junior approves, one senior approves, then the owner manually merges.

Yeah that’s what I think we need to enforce. To answer your question, it was not tagged as AI generated. Frankly, I think we should ban AI-generated code outright, though labeling it as such would be a good compromise.

Pretty much. Someone on our team put out a code review for some new feature and then bounced for a 2 week vacation. One of our junior engineers approved it. Despite the fact that it was in a section of dead code that wasn’t supposed to even be enabled yet, it managed to break our test environment. Took senior engineers a day to figure out how that was even possible before reverting. We had another couple engineers take a look to see what needs to be done to fix the bug. All of them came away with the conclusion that it was 1,000 lines of pure AI-generated slop with no redeemable value. Trying to fix it would take more work than just re-implenting from scratch.

The surprising thing isn’t that physics remain the same from one day to another, it’s that that fact is the reason for conservation of energy. There are lots of different symmetries for the laws of physics: the laws don’t change from one day to another, they don’t change from one part of the universe to the next, and they don’t change based on angles (e.g. if you snapped your fingers and rotated the entire universe by 10 degrees around some arbitrary point, the universe would continue exactly the same as before, just 10 degrees rotated). From Noether’s theorem, you can take any symmetry on the laws of physics, and use that to derive a conservation law. In those examples, that gives you conservation of energy, conservation of momentum, and conservation of angular momentum, respectively.

The Department of Defense DID NOT used to be called the Department of War. Before there was no central department for the entire military. Instead, there was the Department of the Navy and the Department of War (which was for the Army).

By design, other processes cannot inspect what environment variables are running in a container.

That’s not exactly true. If a process is running in a container, and someone is running bash outside of that container, reading that processes environment variables is as simple as “cat /proc/<pid>/environ”. If you meant that someone in one container cannot inspect the environment variables of a process running in a different container, that’s more true. That said, containers should not be considered a security boundary in the same way a hypervisor is.

There are some patterns that emerge when you look at them in other bases. For example, in base 6, the final "digit" (hexit?) is either 1 or 5, (with the exception of the primes 2 and 3). In base 4, the final "digit" is either 1 or 3 (with the exception of the prime number 2). Of course mathematicians generally don't talk about this in terms of their base representation, they usually talk about the primes modulo 6 or the primes modulo 4.

And some of those representations actually do reveal some patterns. For example, an odd prime (so any prime other than 2) p can be written as the sum of two squares p = x^2 + y^2 if and only if p = 1 (mod 4). So those primes that end in 1 in the base 4 representation can be written as the sum of two squares, but the ones that end in 3 cannot. This is called Fermat's theorem on sums of two squares: https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_... .

My guess is that there's a number of different theorems about prime numbers that are phrased in terms of modulo arithmetic or whatever that can be converted into statements about the base representations of primes.

If I had to guess, though, I would guess there isn't a base where the pattern suddenly looks regular. That's very much a guess, but I have a couple data points to support that. The first is Dirichlet's prime number theorem: https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith... . For any coprime integers a and b, the sequence a, a + b, a + 2b, a + 3b, ... contains infinite primes. This seems to imply that primes are, in some sense, evenly distributed across the different possible last "digits" of any base-b representation. There's also the Green-Tao theorem (https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem ) which says that there exist arbitrarily long arithmetic progressions. So for any integer k, there exists some a and b such that a, a + b, a + 2b, ... a + (k-1)b, and a + k*b are all prime! I don't have a good formal argument, but that seems like it would introduce arbitrary "noise" into any proposed pattern of "digits".

Finally, there's the Riemann Hypothesis. This is both my strongest data point and also my weakest data point. There's a deep relationship between the number of primes less than a given number and the zeroes of the Riemann Zeta function. Any pattern on the base-n representation of primes would imply some pattern on the number of primes less than a given number, which would in turn imply some pattern on the zeroes of the Riemann zeta functions. But the Riemann Hypothesis remains unsolved after over 150 years, despite being one of the most-studied problems in number theory. It seems like any pattern in the base-n representation would have meant some pattern in the zeroes of the zeta function, which means we would have made some progress on the Riemann Hypothesis after all this time. I consider this argument both very convincing and not convincing at all, because on one hand I'm relying on the lack of progress of so many people on this problem, which seems convincing, but also maybe it's basically just a logical fallacy, like an appeal to authority.

The AN-TPY/2 radar only sees in a 90 degree arc. Given the primary threat in terms of ballistic missiles is from Iran and maybe Yemen (do the Houthis have ballistic missiles that can range Qatar?) they probably weren’t facing in that direction. I’m not sure how SBIR sensors are managed, but I’m assuming Space Force, and I don’t think there is standard automated data sharing of those findings to THAAD or Patriot batteries in the field. So you would probably be looking at someone in Space Force noticing this and sending some sort of priority alert to everyone in the area. Then the battery crews see the alert and turn their radars around, which takes a second. And then you have to think about launch angles for the missiles. If the battery was stationed a couple kilometers from the intended target, fine. But I’m willing to bet they were probably deployed further away, maybe much further away. I believe they are mostly there to protect the US base there. Quick googling estimate indicates maybe 50km, which is closer than I thought but still not close. All of this also assumes that the missile batteries were fully deployed and on ready alert too, which may not have been true.

I should clarify that it is ENTIRELY possible that you are correct. It is entirely possible the US made a choice to not intercept here. My point is that it is also entirely possible that the US didn’t make a choice. I just don’t see enough clear data to tell. I also think that, even if it was a choice, it was probably more of a split-second decision made by lower-level officers on the ground as it happened, rather than a carefully-considered decision made by national security leadership.

The US probably was able to detect the launch, yes. But would they have assets in place to conduct an intercept in time? Different systems are meant for different targets, and physics put fairly stringent requirements on where an interceptor launcher has to be to protect a specific area. Plus you've got only a few minutes from missile launch to impact. A few minutes to figure out missiles have been launched, confirm it's not a bug or an anomaly, figure out where the target is, get that information from the person monitoring the sensors to command and control plus the missile launch crews, get authority to launch (which depending on the rules of engagement may require going high up the chain of command), get fire control radar tracking the targets (something like SBIR wouldn't be enough, the missile battery needs its own high-quality track), and then launch the interceptor. Also add that people are human beings, and that no one was expecting an eminent strike on Qatar from anyone.

I'm skeptical of that claim. I'm not even sure the Israelis would need refueling assets for this. According to the DoD source in the article, the ALBMs were fired from aircraft over the red sea. That's not far from Israel. Depending on the aircraft used and the exact loadout aerial refueling would probably be unnecessary. If it was necessary, Israel has aerial refueling assets of its own: they were able to conduct a strike campaign against Iran, which is much further away, without foreign tanker support.

It just doesn't make sense to me. This seems well within Israel's own capabilities. Why would they even ask another country for support? Just adds diplomatic complexities and increases the risk of the strike getting leaked.

Each system has its pros and cons, and as a result both methods are widely used. Even the US isn't standardized. The US Air Force I believe largely uses the boom method, while the US Navy largely uses the probe-and-drogue system. Though I believe some aerial tankers are capable of providing fuel via either method.

The strictness of our memory requirements wasn’t made apparent until years later. For background, the application is a system daemon running on end-user servers. So every byte we allocate, every cycle of CPU we use, is a byte or cycle taken away from our customers. We don’t provide firm guarantees or formal SLAs on our memory usage, but we do try to minimize it whenever possible. Because what we don’t want is for someone to upgrade agent versions and suddenly our daemon is using significantly more memory and causing OOMs, or using more CPU and force customers to scale out their fleet. Our p99 cpu and memory usage right now are more or less the same as they were two years ago (RSS is under 40MB last I checked)

So it’s not like we’re running on a machine with only kilobytes of RAM, but we do want to minimize our usage.

No, Go isn’t actually that widely used at my company. The original developers chose Go because they thought it was a good fit for our use case. We were particularly looking for a compiled language that produces binaries with minimal dependencies, didn’t have manual memory management, and was relatively mature (I think Rust was barely 1.0 at the time). We knew we wanted to limit memory usage, but it was more of a “nice to have” than anything else. And Go worked pretty well. It was in production for a couple years before we started getting burnt by these issues. We are looking at porting this to Rust, but that’s a big lift. This is a 50K+ line code base that’s pretty battle tested.