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Sky encrypted phone.
by jim bell 19 Mar '21

19 Mar '21

VICE: Encrypted Phone Firm 'Sky': Someone Sold Compromised Versions of Our App. https://www.vice.com/en/article/epd3km/sky-ecc-hacked-fake-app

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ASSANGE washes his lily hands of Snowden, Iraq, refugees & Whistleblowers
by professor rat 19 Mar '21

19 Mar '21

"... Mr Assange agreed that some level of privacy was necessary for the successful operation of the military ..." http://www.abc.net.au/radionational/programs/drive/assange-i-can-rule-from-… JULIAN " ‘[The military] protects the sovereignty of Australia. It protects the independence of Australia.' ASSANGE July, 2013

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ping
by Punk-BatSoup-Stasi 2.0 19 Mar '21

19 Mar '21

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small rehash of part of modern cointelpro/disruption tactics from activism
by Karl 19 Mar '21

19 Mar '21

Private Military Corporations (PMCs) are essentially mercenaries acting largely outside of government regulation or democratic control. They are hired by private corporations to assist in their interests and act as for-hire businesses with few or no ethical considerations. Some examples of these corporations are TigerSwan, Triple Canopy, and STRATFOR. PMCs are often staffed with U.S. military veterans, and employ counterinsurgency techniques and skills honed during the invasions of Iraq and Afghanistan, or other military operations. And in many cases, these PMCs collaborate with public law enforcement agencies to share information, such that law enforcement is essentially acting as a private contractor <https://dgrnewsservice.us13.list-manage.com/track/click?u=89a226816d8aa73fa…> for a corporation. Disruption Tactics Used by Corporate Goon Squads PMCs can be expected to deploy four basic tactics. 1. * Intelligence Gathering* First, they will attempt to gather as much information on protesters as possible. This begins with what is called OSINT — Open Source Intelligence. This simply means combing through open records on the internet: Googling names, scrolling through social media profiles and groups, and compiling information that is publicly available for anyone who cares to look. Other methods of information gathering are more active, and include physical surveillance (such as flying a helicopter overhead, as occurred today), signals intelligence (attempting to capture cell phone calls, emails, texts, and website traffic using a device like a Stingray also known as an IMSI catcher), and infiltration or human intelligence (HUMINT). This last is perhaps the most important, the most dangerous, and the most difficult to combat. 1. * Disruption* Second, they will attempt to disrupt the protest. This is often done by using the classic tactics of COINTELPRO to plant rumors, false information, and foment infighting to weaken opposition. During the protests against the Dakota Access Pipeline, one TigerSwan infiltrator <https://dgrnewsservice.us13.list-manage.com/track/click?u=89a226816d8aa73fa…> working inside the protest camps wrote to his team that “I need you guys to start looking at the activists in your area and see if there are individuals who are vulnerable. They’re broke, always talking about needing gas money or whatever. Maybe they’re disillusioned, depressed a little. Life is fucking them over. We can buy them a bus ticket to any camp they want if they’re willing to provide intel. We win no matter what. If they agree to inform for pay, we get intel. If they tell our pitchman to go f*** himself/herself, the activist will start wondering who did take the money and it’ll cause conflict within the activist groups and it won’t cost us anything.” In 2013, there was a leak of documents from the private intelligence company STRATFOR, which has worked for the American Petroleum Institute, Dow Chemical, Northrup Grumman, Coca Cola, and so on. The leaked documents revealed one part of STRATFOR’s strategy for fighting social movements. The document proposes dividing activists into four groups, then exploiting their differences to fracture movements. “Radicals, idealists, realists and opportunists [are the four categories],” the leaked documents state. “The Opportunists are in it for themselves and can be pulled away for their own self-interest. The Realists can be convinced that transformative change is not possible and we must settle for what is possible. Idealists can be convinced they have the facts wrong and pulled to the Realist camp. Radicals, who see the system as corrupt and needing transformation, need to be isolated and discredited, using false charges to assassinate their character is a common tactic.” As I will discuss later on, solidarity and movement culture is the best way to push back against these methods. Other examples of infiltration and disruption have often focused on: - Increasing tensions around racist or sexist behavior - Targeting individuals with drug or alcohol addictions to become informants - Using sex appeal and relationship building to get information - Acting as an “agent provocateur” to encourage protesters to become violent, even to the point of supplying them with bombs, in order to secure arrests - Spreading rumors about inappropriate behavior to sew discord and mistrust 1. * Intimidation* The third tactic used by these companies is intimidation. They will use fear and paranoia as a deliberate form of psychological warfare. This can include anonymous threats, shows of force, visible surveillance, and so on. 1. * Violence* When other methods fail, PMCs and public law enforcement will ultimately resort to direct violence, as we have seen with Standing Rock and many other protest movements. As I have written before, colonial states enforce their resource extraction regimes with force, and we should disabuse ourselves of notions to the contrary. Vigilante violence is also always a concern. When people seek to defend land from destruction, men with guns are usually dispatched to arrest them, remove them from the site, and lock them in cages. How to Resist Against Surveillance and Repression There are specific techniques we can deploy to protect ourselves, and by extension, protect the land at Thacker Pass. These techniques are called “security culture.” Security culture is a set of practices and attitudes designed to increase the safety of political communities. These guidelines are created based on recent and historic state repression, and help to reduce paranoia and increase effectiveness. Security culture cannot keep us 100% safe, all the time. There is risk in political action. But it helps us manage risks that do exist, and take calculated risks when necessary to achieve our goals. The first rule of security culture is this: be cautious, but do not live in fear. We cannot let their intimidation be effective. Creating paranoia is a key goal for PMCs and other repressive organizations. When they make us so paranoid we no longer take action, reach out to potential allies, or plan and carry out our campaigns, they win using only the techniques of psychological warfare. When we are fighting to protect the land and water, we are doing something righteous, and we should be proud and stand tall while we do this work. The second rule of security culture is that solidarity is how we overcome paranoia, snitchjacketing, and rumor-spreading. We must act with principles and in a deeply ethical and honorable way. Work to build alliances, friendships, and trust—while maintaining good boundaries and holding people accountable. This is the foundation of a good culture. In regards to infiltration, security culture recommends the following: 1. It’s not safe nor a good idea to generally speculate or accuse people of being infiltrators. This is a typical tactic that infiltrators use to shut movements down. 2. Paranoia can cause destructive behavior. 3. Making false/uncertain accusations is dangerous: this is called “bad-jacketing” or “snitch-jacketing.” 4. Build relationships deliberately, and build trust slowly. Do not share sensitive information with people who don’t need to know it. There is a fine line between promoting a campaign and sharing information that could put someone at risk. 5. Good security culture focuses on *identifying and stopping bad behavior.* 6. Do not talk to police or law enforcement unless you are a designated liaison. Secure communications are an important part of security culture. Here are some basic recommendations to secure your communications. 1. Email, phone calls, social media, and text messages are inherently insecure. Nothing sensitive should be discussed using these platforms. 2. Preferably, use modern secure messaging apps such as Signal, Wire, or Session. These apps are free and easy to use. 3. We recommend setting up and using a VPN for all your internet access needs at camp. ProtonVPN and Firefox VPN are two reputable providers. These tools are easy to use after a brief initial setup, and only cost a small amount. Invest in security. We must also remember that secure communications aren’t a magic bullet. If you’re communicating with someone who decides to share your private message, it’s no longer private. Use common sense and consider trust when using secure communications tools. Security culture also warns us not talk about some sensitive issues, including: - Your or someone else’s participation in illegal action. - Someone else’s advocacy for such actions. - Your or someone else’s plans for a future illegal action. - Don’t talk about illegal actions in terms of specific times, people, places, etc. Note: Nonviolent civil disobedience is illegal, but can sometimes be discussed openly. In general, the specifics of nonviolent civil disobedience should be discussed only with people who will be involved in the action or those doing support work for them. It’s still acceptable (even encouraged) to speak out generally in support of monkeywrenching and all forms of resistance as long as you don’t mention specific places, people, times, etc. Conclusion Security is a very important topic, but is challenging. There are so many potential threats, and we are not used to acting in a secure way. That’s why we are working to create a “security culture”—so that our communities of resistance are always considering security, assessing threats, studying our opposition, and creating countermeasures to their methods. Most importantly, do not let this scare you, and do not be overwhelmed. Simply take one security measure at a time, begin to study it, and then implement better protocols one by one. We use the term “security culture” because security is a mindset that should be developed and shared.

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Ellsberg Snowden Assange: Blowing Lies of the Espionage Act
by grarpamp 19 Mar '21

19 Mar '21

Daniel Ellsberg Talks About Whistleblowing, The Pervasiveness Of Official Lies, And The Dangers Of The Espionage Act https://www.youtube.com/watch?v=tKGPYzLTvFY Ellsberg: Being Used to Smear https://taibbi.substack.com/p/daniel-ellsberg-talks-about-whistleblowing https://www.history.com/topics/vietnam-war/pentagon-papers Authored by Matt Taibbi via TK News, “On Tuesday morning, August 4th, 1964,” writes Daniel Ellsberg in Secrets: A Memoir of Vietnam and the Pentagon Papers, “a courier came in my out office with an urgent cable for my boss. He had been running.” A former Marine with a PhD from Harvard in Decision Theory, Ellsberg had joined the Pentagon as special assistant to Assistant Secretary of Defense John McNaughton, who himself was perhaps the closest advisor to Secretary Robert McNamara. Ellsberg, in other words, was the right hand of the right hand, of the man who would become known as the chief architect of the Vietnam War. Ellsberg’s first day on August 4th, 1964 proved to be a historic one. His boss McNaughton was down the hall with McNamara, so the panting courier handed Ellsberg the note and left. He opened it and found it was from Captain John J. Herrick, the commodore of a two-destroyer flotilla in the Gulf of Tonkin, off North Vietnam in the South China sea. Officially, the United States was not yet engaged in full-fledged military operations in Indochina. Daniel Ellsberg: “We could be East Germany in weeks, in a month. Huge concentration camps and so forth…” Herrick said he was under attack by North Vietnamese patrol boats, and had opened fire in return. He was 60 miles from the coast, in international waters. The sonar operators on the Destroyers Maddox and Turner Joy, Maddox said, each heard torpedoes in the water. Ten minutes later, the courier returned with a new note. “Am under continuous torpedo attack,” he wrote, about an encounter that was taking place in total darkness. For some time after, cables came in quick succession, as Ellsberg guessed Herrick was dictating from the bridge in between trying to maneuver his ships. “Torpedoes missed. Another fired at us,” read one. “Four torpedoes in water,” read a second. “Five torpedoes in water… Have successfully avoided at least six torpedoes…” According to Herrick, at least one attacking boat had been sunk. The action went on for two long hours, before suddenly the stream of messages cut short. “Then, suddenly, an hour later,” Ellberg wrote, “a message arrived that took back, not quite all of it, but enough to put the rest of it in question.” The courier came in running again, handing him a cable with the highest clearance and urgency [emphasis mine]: Review of action makes any reported contacts and torpedoes fired appear doubtful. Freak weather effects on radar and overeager sonar men may have accounted for many reports. No actual visual sightings by Maddox. Suggest complete evaluation before any further action taken… It was a little after 2 p.m., Washington time. Ellsberg was dumbfounded by the latest communications. “In my mind, these messages erased the impact of the two-hour-long live drama that we had been following. This new information was a cold bath.” Herrick later sent another cable: “Details of action present a confusing picture, although certain original ambush bona fide.” Ellsberg was now unsure of how Herrick was so sure, given that he hadn’t seen anything and was acknowledging, among other things, that one sonar man was hearing his own ship’s propeller. “It seemed almost certain there had been no attack,” Ellsberg wrote, certain the proper course was to wait to see what actually happened before acting. Things didn’t go that way. Senior military officials scrambled to put together an immediate retaliatory airstrike. President Lyndon Johnson was so anxious not only to strike back, but to brief the public about doing it, that he asked the Pentagon’s permission to go on TV with details before the planes even reached Vietnam. LBJ was on the air by 11:37 p.m. that night, telling the American people that “hostile vessels attacking two U.S. destroyers with torpedoes” constituted “open aggression on the high seas against the United States of America.” McNamara gave subsequent pressers in which he described “unprovoked” attacks of U.S. vessels on “routine patrols” in “international waters.” They described the evidence for Vietnamese aggression as “unequivocal.” By the end of Ellsberg’s first day, he knew every single one of these claims was a lie. The two destroyers were on a special mission, penetrating deep into North Vietnamese waters and engaging in sabotage raids. In top-secret testimony to congress in the two days after the August 4th incident, McNamara and Secretary of State Dean Rusk told congressional leaders the U.S. had nothing to do with the raids, which were entirely South Vietnamese operations. Ellsberg soon learned this was a lie, too, that the personnel on the ships had been chosen by the CIA and that the operations were run jointly by the agency and the Navy. “Each of these assertions,” Ellsberg would later write, “was false.” You can still go back and look to see how these lies were reported with complete credulity and never corrected: Ellsberg became famous years later for shepherding to the public a wealth of secret documents about the ugly history of failure, brutality, and ignorance in the Vietnam War, collectively known as the Pentagon Papers. He is America’s most famous whistleblower, a figure who single-handedly triggered a major constitutional crisis when the government of Richard Nixon tried to block publication of his material. However, Ellsberg has remained an important figure in American culture and politics precisely because so little has changed since the events of the fifties, sixties, and seventies he described in such vivid detail. In the Useful Idiots interview below, Ellsberg points out the similarities between Vietnam and our current policies in various countries around the world. He says our leaders are worried about “regime change in Washington,” which they believe would occur if they left other countries’ oil in the ground, or “stopped killing Afghans.” More than anything, however, Ellsberg is an expert on the role of secrecy in American life. Both in his books and in his interview with Useful Idiots, he describes military and executive branch officials who don’t even figure “truth” as a variable in their calculations, since it’s irrelevant to what they tell the world. He arrived in Washington believing the commonly held notion that nothing in the capital stays secret for long. Soon he learned that it’s actually quite easy to keep secrets. Ellsberg described a vicious cycle, in which leaders lie pervasively, then learn to have so much contempt for the public that swallows those lies, that they feel justified in lying more. “My awareness of how easily Congress, the public, and journalists were fooled and misled contributed to a lack of respect for them,” he wrote. “That, in turn, made it easier to accept practices of deception,” and “their resulting ignorance made it all the more obvious that they must leave these problems to us.” Ellsberg is adamant that our military and intelligence services don’t learn from even the bloodiest failures. However, when asked in the Useful Idiots if they’d at least learned something in a negative sense — like how to deal with whistleblowers and shut off pictures of war deaths — he concurred, explaining that he himself had been used as propaganda. “It is now accepted that somebody can be a good whistleblower, and that’s Daniel Ellsberg,” he says, “in contrast with Chelsea Manning and Ed Snowden. The appreciation that I’ve been getting since 2010, I can date very simply to the need to denigrate Chelsea Manning.” He went on to describe a New Yorker piece written by Malcolm Gladwell that ripped Ed Snowden in comparison to him, Ellsberg, among other things quoting an analyst who wondered if Snowden “may have been the dupe of a foreign-intelligence service.” Ellsberg wrote a letter to the New Yorker calling the contrast ridiculous, and, he tells us, “They never published it.” Overall, Ellsberg’s takes on nuclear safety, the implications of the use of the Espionage Act in the Julian Assange case, and continued misuse of secrecy and hyper-aggressive foreign policy in places like Afghanistan and Syria, still resonate. The most powerful part of his interview regarded the power of the secret state in modern America. “They know where we are, they know our names, they know from our iPhones if we're on our way to the grocery store or not,” he said. “We could be East Germany in weeks. In a month.” The last portion of the Useful Idiots episode: Excerpt from the interview: Matt Taibbi: What you saw in Vietnam is similar to what people saw in Iraq, and then Afghanistan. What's the mentality that continues to think that these same kinds of policies will work, and why can't they get out of that mentality? Ellsberg: You have to ask, who is it who actually bears the cost of these and who doesn't? Any of these wars were not bad for the people making weapons, and it's not only them. It's the banks that finance them and it's the congresspeople who benefit, as I keep saying, from the donations and the jobs and so forth. They did fine… Are we actually going to get out of Afghanistan? It's scheduled, by Trump of all people, for May. Okay, that's very close. Is that going to happen? Let's see. Certainly not for sure… If we don't get out now, there is no reason why it will look different two years from now, five years from now, 10 years from now. We'll still be killing Afghans and losing very few Americans, because it's all in the air and some special forces going to unarmed villages and whatnot. So very few American casualties, air power, a lot of things. The American public can live with that for a long time. They have lived for 20 years with that. Could be another 20 years. Katie Halper: Can you talk about the role of the media in America’s aggressive foreign policy? Ellsberg: With the Gulf of Tonkin, the Times did not say, "Here's what we said at the time." The Times did not go back and say, "Here's who lied to us. Here's how we were lied to. Here's how gullible we were. Here's the pattern of deception." No, that would blame themselves. They didn't need that, so they didn't do it… In short, I think nations and institutions, we talk about why don't they learn, learning is not what they do, because learning involves seeing prior errors that you haven't met. Errors are an occasion for blame, for losing jobs, for being criticized, and they don't do that. Matt Taibbi: Well, sometimes they learn in a negative way though, don't they? Do you ever think that the way they dealt with Snowden, and to a lesser extent Julian Assange and some other whistleblowers, was about making sure that there was never going to be a Daniel Ellsberg again who would live on and be a hero in the public consciousness? Ellsberg: I misspoke when I say they don't do any learning… Definitely, they do learn. Katie Halper: They don't become more moral, though. Ellsberg: It's not as though they learn how to meet human values or improve human welfare in the world. That's not what they're into… But in terms of how can we get away with it better, they do learn…. They've learned to wield the Espionage Act, to criminalize whistleblowing much more than before. You said they didn't want any more Ellsbergs. Well, obviously, they did get Chelsea Manning, they did get Snowden. Chelsea was 39 years after the Pentagon Papers. The Pentagon Papers did have an effect, as you say, on people's understanding of the war. It didn't end the war, but it did affect people's attitudes. And really, it kept us out of more Vietnams for a couple of decades… The NSA did not do surveillance on American citizens without a warrant for about 25 years or so after, so that was a change. But then 9/11 comes along, and it’s Constitution be damned. Since then, this is 20 years ago, we've had total surveillance of everybody, totally unconstitutionally. It's created a situation where we're not a police state, but we could be a police state almost from one day to the next, if they act on all the information they have now about people who give them any trouble or people who protest. They know where we are, they know our names, they know from our iPhones if we're on our way to the grocery store or not. But they haven't acted on that to put people in camps yet. They could do it. We could be East Germany in weeks, in a month. Huge concentration camps and so forth. Matt Taibbi: Why aren’t more journalists worried about the use of the Espionage Act in the Julian Assange case? Ellsberg: Because they have never been tried before. This is a first, so they thought they were immune… I've been saying for 40 years now, 50 years, I've been saying to journalists and judges, the wording of that law applies to you as well as your sources… It is now accepted that somebody can be a good whistleblower, and that's Daniel Ellsberg, in contrast to Chelsea Manning and Ed Snowden. The appreciation I've been getting since 2010, I can date very simply to the need to denigrate Chelsea Manning… This contrast is used all the time. So I'm appreciated in order to say, "Ah, but there were bad whistleblowers like Snowden or Assange." And a lot of people do that.

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Cutting Edge Automation
by Karl 19 Mar '21

19 Mar '21

A recent small innovation in artificial "intelligence" was the publication of a system called "cibi", which autonomously wrote code in the dainty language "brainfuck", in a way that was competitive with openai's environment benchmarks: https://arxiv.org/pdf/2101.09571v3.pdf The code unfortunately has been removed from github: https://github.com/vadim0x60/cibi However, the author says it will be back soon: https://github.com/vadim0x60/cibi-experiments/issues/4 Meanwhile, if you want to write a new AI virus or try to solve automatic world peace in a way that actually uses transparent respectful communication instead of covert direct result, there are many other reinforcement learning approaches out there ... In the field of machine learning, a bit ago a group called eleuther-ai open sourced an alternative to the expensive private text generation algorithm GPT-3: https://www.eleuther.ai/ The license they chose is MIT. And cell phones are making daily strides towards turning into that future with end-user apps, if you check the right places in the commercial app stores. I ran into an early reinforcement algorithm on facebook while politically targeted and now copy traumatic echoes of its patterns with my behavior everywhere. It was mostly about covering itself up. Apparently it's a big thing to have unsupervised reinforcement learning algorithms "accidentally" mind control people; it's like a serious problem actually being studied, if you look into the field of alignment. It's one of many problems that are being cast as pretty hard to mathematically avoid.

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The Mysterious Math of Perfection
by coderman 18 Mar '21

18 Mar '21

https://www.quantamagazine.org/the-mysterious-math-of-perfect-numbers-20210… The Mysterious Math of Perfection Enter the world of perfect numbers and explore the mystery mathematicians have spent thousands of years trying to solve. [BIG MOUTH](https://www.behance.net/BIGMOUTH) for Quanta Magazine Patrick Honner Contributing Columnist --------------------------------------------------------------- March 15, 2021 --------------------------------------------------------------- Mona Lisa’s smile. Mary Lou Retton’s Olympic vault. Mariah Carey’s musical pitch. All are considered perfect. So are the numbers 6 and 28. With feats of artistry and athleticism, perfection lies in the eye of the beholder. But for numbers, perfection is mathematically defined. “Perfect numbers” are equal to the sum of their “proper” divisors (positive integers that divide a number evenly, not counting itself). For example, 6 = 3 + 2 + 1, and 28 = 14 + 7 + 4 + 2 + 1. While these mathematical curiosities are about as likely to grace the walls of the Louvre as they are to perform a twisting layout back somersault, they do offer something irresistible: a perfect mystery. Euclid laid out the basics of perfect numbers over 2,000 years ago, and he knew that the first four perfect numbers were 6, 28, 496 and 8,128. Since then, many more perfect numbers have been discovered. But, curiously, they’re all even. No one has been able to find an odd perfect number, and after thousands of years of unsuccessful searching, it might be tempting to conclude that odd perfect numbers don’t exist. But mathematicians haven’t been able to prove that either. How is it that we can know so much about even perfect numbers without being able to answer the simplest question about an odd one? And how are modern mathematicians trying to resolve this ancient question? Our exploration of mathematical perfection begins with divisors. We know that 6 is a divisor of 12 since 126 = 2, and we know 25 is a divisor of 100 since10025 = 4. As we said, we know a number is perfect when it is equal to the sum of its proper divisors — those divisors that are less than the number itself. We can also define a number as perfect when the sum of all its divisors, proper and improper, is twice the number. This works out because the only improper divisor of a number is the number itself. We see that 28 is still perfect by this definition: Its proper divisors are 1, 2, 4, 7 and 14, its improper divisor is 28, and the sum of all its divisors, 1 + 2 + 4 + 7 + 14 + 28, is 56, which is 2 × 28. Including the improper divisor in the sum is convenient for some of the algebra we’ll do with perfect numbers, as we’ll see shortly. Quantized Academy Patrick Honner, a nationally recognized high school teacher from Brooklyn, New York, introduces basic concepts from the latest mathematical research. --------------------------------------------------------------- When working with perfect numbers we end up saying “the sum of the divisors of a number” a lot, so mathematicians made things easier by turning this into a function. We’ll define σ(n), or “sigma of n,” to be the sum of the divisors of n. We already know that σ(28) = 56. Some other examples: σ(1) = 1, σ(6) = 1 + 2 + 3 + 6 = 12, and σ(10) = 1 + 2 + 5 + 10 = 18. Notice that 6 is a perfect number, since σ(6) = 2 × 6, but 1 and 10 are not. As we’ll see, this function σ has some special properties that are perfect for studying perfect numbers. So we’ve got our basic definition of perfect numbers and a new mathematical tool to help us find them. Where should we start looking? We’ll start where mathematicians always start when studying numbers and their patterns: the primes. A prime number, by definition, is divisible only by itself and 1. This makes computing σ for a prime number quite easy: σ(2) = 1 + 2 = 3, σ(3) = 1 + 3 = 4, σ(5) = 1 + 5 = 6, and σ(7) = 1 + 7 = 8. In general, for any prime number p, σ(p) = 1 + p. Could a prime number be perfect? Only if σ(p) = 1 + p = 2p. A little bit of algebra tells us this will be true whenever p = 1, but since prime numbers are greater than 1 by definition, no prime could be perfect. So we know primes can’t be perfect. Where should we look next? Powers of primes — numbers like 24, 53 or 1136 — are a good next step, as their divisors are easy to organize. Consider a prime power like 16, or 24. The only divisors of 24 are the powers of 2 up to 24: 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16. So σ(24) can be computed like this: σ(24) = 1 + 2 + 22 + 23 + 24 = 1 + 2 + 4 + 8 + 16 = 31. And this generalizes. For any prime power pn σ(pn) = 1 + p + p2 + p3 + … + pn. This becomes even easier if we use a formula from algebra class. Notice that each of the terms being added up in σ(pn) is p times the previous term. This makes this a so-called geometric series, and there’s a nice formula for the sum of a geometric series: 1 + p + p2 + p3 +…+ pn = pn+1−1p−1 . (For more on this amazing formula, see the exercises at the end of the column.) Thanks to the geometric series formula, we don’t have to list out all the divisors of pn to compute σ(pn). We can just use the formula: σ(pn) = 1 + p + p2+ p3+…+ pn = pn+1−1p−1 . For example, we’ve already seen that σ(24) = 1 + 2 + 22 + 23 + 24 = 25−12−1 =32−11 = 31. And we can compute the sum of the divisors of other prime powers by just plugging them into the formula: σ(27) = σ(33) = 34−13−1 =81−12 = 40 and σ(121) = σ(112) = 113−111−1 =1331−110 = 133. Notice that none of these prime powers satisfies the condition for being perfect: σ(24) ≠ 2 × 24, σ(33) ≠ 2 × 33, and σ(112) ≠ 2 × 112. In fact, no prime power can be perfect. To get a perfect number we need σ(pn) = 2pn, which would mean: 1 + p + p2 + p3 + … + pn-1 + pn = 2pn. We can subtract pn from both sides of the equation to give us: 1 + p + p2 + p3 + … + pn-1 = pn. Now we’ll use the geometric series formula on the left-hand side of this equation: Since 1 + p + p2 + p3 + … + pn-1 = pn−1p−1 we get pn−1p−1 = pn. We need this to be true for pn to be perfect. But notice that pn – 1 is smaller than pn, and dividing pn – 1 by p – 1 will make it even smaller, so pn−1p−1 < pn. And thus no prime power pn can be perfect. So no perfect primes, and no perfect prime powers. What can be perfect? Well, we know 28 is perfect, and it is the product of two distinct prime powers: 28 = 22 × 7. Any number that isn’t a prime or a prime power can be written as the product of distinct prime powers like this. And these factorizations, together with a special property of the function σ, can help us determine if a number is perfect. We already know that σ(28) = 1 + 2 + 4 + 7 + 14 + 28, but let’s take a closer look at that sum. Notice that each of the last three numbers is a multiple of 7: We can factor out that 7 to reveal some hidden structure: σ(28) = (1 + 2 + 4) + 7 × (1 + 2 + 4). And with some more clever factoring with the distributive property, we can write σ(28) = (1 + 2 + 4)(1 + 7). This doesn’t tell us anything we didn’t already know: σ(28) = (1 + 2 + 4)(1 + 7) = 7 × 8 = 56, which confirms that 28 is perfect. But there’s something important hiding inside that multiplication: σ(28)=(1+2+4)(1+7)=(1+21+22)(1+71). Those expressions in parentheses look familiar: 1 + 21 + 22 = σ(22), and 1 + 71 = σ(7). This means that we can actually write σ(28) = σ(22)σ(7). To compute σ(28) = σ(22 × 7) we can actually compute σ(22) and σ(7) and multiply them. This is a surprise, and it’s true in general: Any time you factor a number into primes like this you can use this shortcut to compute σ. For example, since 100 = 22×52, we can compute σ(100) like this: σ(100) = σ(22)σ(52) = (1 + 2 + 4)(1 + 5 + 25) = 7 × 31 = 217, which is a bit easier than listing all nine divisors of 100 and adding them up. Why does this work? Well, the divisors of a number come from its prime factors. Consider 28 again, which is the product of 22 and 7, and think about the multiplication table below: x 1 2 4 1 7 Along the top are the powers of 2 that evenly divide 28, and down the side are the powers of 7 that evenly divide 28. Notice what happens when we fill out this multiplication table. x 1 2 4 1 1 2 4 7 7 14 28 We get all the divisors of 28. That’s because every divisor of 28 is a combination of divisors of 22 and 7, the prime powers that appear in the factorization of 28. Now compare the multiplication table with the expression (1 + 2 + 4)(1 + 7). When we multiply this out using the distributive property, this also produces all the divisors of 28 and then adds them up: (1 + 2 + 4)(1 + 7) = 1 × 1 + 2 × 1 + 4 × 1 + 7 × 1 + 7 × 2 + 7 × 4. In other words, (1 + 2 + 4)(1 + 7) is exactly σ(28). But (1 + 2 + 4)(1 + 7) is also σ(22)σ(7). So σ(22)σ(7) = σ(28). This example demonstrates a very useful fact about σ: In the language of number theory, this function is “multiplicative.” That means that σ(ab) = σ(a)σ(b) whenever the numbers a and b are “relatively prime,” meaning they have no factors in common. This is the special feature of σ that’s perfect for helping us study perfect numbers. Euclid used this fact 2,000 years ago to create a formula for finding perfect numbers, with help from a special kind of prime and a clever argument about products and divisors. In doing so, he took the first step toward determining what every even perfect number has to look like. Let’s see how he did it. First, notice that for any power of 2 we have σ(2k) = 2k+1−12−1 = 2k+1 – 1. This is a consequence of the geometric series formula we discussed earlier. Now consider the following thought experiment: What if 2k+1 – 1 is prime? Well, since σ(p) = 1 + p for any prime, we know that σ(2k+1 – 1) = 1 + 2k+1 – 1 = 2k+1. And notice that 2k+1 is exactly twice 2k, because of the law of exponents that says 2 × 2k = 2k+1. So we have the following two interesting relationships between the numbers 2k and 2k+1 – 1: σ(2k) = 2k+1 – 1 and σ(2k+1 – 1) = 2k+1 = 2 × 2k. Euclid noticed a clever way to leverage these relationships: He put the two numbers together to make the number M = 2k × (2k+1 – 1), and as long as (2k+1 – 1) is prime, this number is perfect! To see this, we’ll compute σ(M) and show it is equal to 2M. First, notice that 2k+1 – 1 is one less than an even number so it must be odd. This means 2k+1 – 1 is not divisible by 2. But 2k is only divisible by powers of 2. So 2k and 2k+1 – 1 have no common factors and are thus relatively prime. This allows us to use the multiplicative property of σ: We already know that σ(2k) = 2k+1 – 1 and σ(2k+1 – 1) = 2k+1 = 2 × 2k, so we can find σ(M): σ(M)=σ(2k×(2k+1−1))=σ(2k)σ(2k+1−1)=(2k+1−1)(2×2k)=(2×2k)(2k+1−1)=2(2k×(2k+1−1))=2(M). So M = 2k × (2k+1 – 1) is perfect, as claimed. Keep in mind that this relies on the assumption that the number 2k+1 – 1 is prime. These numbers are called Mersenne primes, and you might have heard of them because of the Great Internet Mersenne Prime Search ([GIMPS](https://www.mersenne.org/)) a collaborative online computing effort to find huge Mersenne primes. Anytime you hear about the discovery of a new largest prime number, it’s probably the result of GIMPS. And thanks to Euclid’s proof, anytime a new Mersenne prime is discovered, a new perfect number is discovered as well. For example, 25 – 1 = 31 is a Mersenne prime, and so 24(25-1) = 16 × 31 = 496 is a perfect number. Also, 22 – 1 = 3 is a Mersenne prime, so 21(22 – 1) = 2 × 3 = 6 is perfect. And 23 – 1 = 7 is a Mersenne prime, so 22(23 – 1) = 4 × 7 = 28 is perfect. You may have noticed that all these perfect numbers are even. This makes sense, because as long as k > 0, the number 2k × (2k+1 – 1) will be even. (And if k = 0 then 2k+1 – 1 is 1, which is not a prime.) You may also have noticed that all the perfect numbers we’ve discussed so far seem to involve Mersenne primes. That’s no coincidence: 2,000 years after Euclid showed that this formula generates perfect numbers, Leonhard Euler proved that this is the only way to get even perfect numbers. But the question of what odd perfect numbers might be like (if they exist) remained open. And it remains open today. Although they can’t find one, mathematicians have a lot of information about what a hypothetical odd perfect number might look like. It can’t be divisible by 105. It would have to have at least nine distinct prime factors, the second-largest of which would have to be greater than 10,000. And it would have to have a remainder of 1 when divided by 12 or a remainder of 9 when divided by 36. It might seem strange to prove results about numbers that might not even exist. But every new rule narrows the search a little more. And if they’re lucky, mathematicians might just prove that odd perfect numbers have to satisfy two incompatible criteria, which would prove once and for all that no odd perfect number exists. On the hunt for incompatible criteria, mathematicians have even started [looking at numbers that aren’t quite perfect](https://www.quantamagazine.org/mathematicians-open-a-new-front-on-…. A “spoof perfect number” is a number that looks perfect if you pretend one of its non-prime factors is actually prime. For example, 60, the product of 3, 4 and 5, can be considered “spoof perfect”: If you pretend that the 4 in its factorization is a prime, then the shortcuts we developed for σ give us (1+3)(1+4)(1+5) = 4 × 5 × 6 = 120. If σ(60) equaled 120, then 60 would be perfect. Of course, σ(60) doesn’t actually equal 120, but it looks like it if we pretend 4 is a prime. That’s what makes it a spoof. These spoofs are like generalizations of perfect numbers, and so anything true about a spoof would have to be true about a perfect number as well. Understanding odd spoofs would be especially useful, since any rule discovered for odd spoofs could be added to the existing rules for odd perfect numbers, increasing the chances of finding contradictory criteria and tightening the overall search space. Related: --------------------------------------------------------------- - [Mathematicians Open a New Front on an Ancient Number Problem](https://www.quantamagazine.org/mathematicians-open-a-new-front-on-… - [The Simple Math Problem We Still Can’t Solve](https://www.quantamagazine.org/why-mathematicians-still-cant-solve-t… - [To Win This Numbers Game, Learn to Avoid Math Patterns](https://www.quantamagazine.org/to-win-this-numbers-game-learn-to-… René Descartes, yet another famous mathematician drawn into the mystery of perfect numbers, discovered the first odd spoof perfect number, and he challenged mathematicians to find others. In taking up the challenge, mathematicians have broadened the concept of a spoof and have discovered a new class of numbers to study. For the most part, investigations into these spoof perfect numbers are done simply for the joy of mathematical exploration. But perhaps something we learn about spoofs will help us prove that actual odd perfect numbers can’t exist, or perhaps it will lead us to one. It may seem strange to spend thousands of years hunting for numbers with curious properties, proving theorems about objects that might not even exist, and inventing new and even stranger worlds of numbers to explore. But to a mathematician, it makes perfect sense. Exercises 1. A number is called “abundant” if it is less than the sum of its proper divisors. For example, 36 is abundant, since 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55, which is greater than 36. What’s the smallest abundant number? 2. A number is “deficient” if it is greater than the sum of its proper divisors. For example, 35 is deficient, since 1 + 5 + 7 = 13, which is less than 35. Are prime powers deficient or abundant? 3. Use the distributive property to multiply (p – 1)(1 + p + p2 + p3 + … + pn) and simplify. 4. Suppose N is an odd number and assume that M = 2N and is perfect. Show that N must be equal to 3.

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ExtremeTech: The Semiconductor Shortage Comes for Qualcomm, Sweeps Phone Industry
by jim bell 16 Mar '21

16 Mar '21

ExtremeTech: The Semiconductor Shortage Comes for Qualcomm, Sweeps Phone Industry. https://www.extremetech.com/extreme/320817-the-semiconductor-shortage-comes…

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Inverse: SpaceX Starlink coverage map: When will it be available in my area?
by jim bell 16 Mar '21

16 Mar '21

Inverse: SpaceX Starlink coverage map: When will it be available in my area?. https://www.inverse.com/innovation/spacex-starlink-when-will-it-be-availabl…

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Carpetbagger invasion
by professor rat 16 Mar '21

16 Mar '21

ZKSwap to Deploy on BSC, Issue 1,993 Cypherpunk NFTs The company will issue 1,993 BSC versions of the ZKSwap-exclusive NFTs to commemorate the first mass media discussion of the Cypherpunk Movement in 1993. https://chainbulletin.com/zkswap-to-deploy-on-bsc-issue-1993-cypherpunk-nft…

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