Month: January 2018

Largest prime number discovered with more than 23m digits

With nearly one million more digits than the previous record holder, the new largest prime number is the 50th rare Mersenne prime ever to be discovered

At more than 23m digits long, the number is something of a beast. But for mathematicians, the latest discovery from a global gang of enthusiasts is a thing of beauty: the largest prime number ever found.

Known simply as M77232917, the figure is arrived at by calculating two to the power of 77,232,917 and subtracting one, leaving a gargantuan string of 23,249,425 digits. The result is nearly one million digits longer than the previous record holder discovered in January 2016.

The number belongs to a rare group of so-called Mersenne prime numbers, named after the 17th century French monk Marin Mersenne. Like any prime number, a Mersenne prime is divisible only by itself and one, but is derived by multiplying twos together over and over before taking away one. The previous record-holding number was the 49th Mersenne prime ever found, making the new one the 50th.

Im very surprised it was found this quickly; we expected it to take longer, said Chris Caldwell, a professor of mathematics who runs a website on the largest prime numbers at the University of Tennessee at Martin. Its like finding dead cats on the road. You dont expect to find two so close to one another.

The new prime number was originally found on Boxing Day by the Great Internet Mersenne Prime Search (Gimps) collaboration which harnesses the number-crunching power of volunteers computers all over the world. In the days after, four more computers sporting different hardware and software were set the task of verifying the discovery. Those computers confirmed the result, taking between 34 and 82 hours each.

To find M77232917 in the first place took six full days of nonstop computing on a PC owned by Jonathan Pace, a 51-year old electrical engineer from Germantown, Tennessee. It is the first prime that Paces computer has churned out in 14 years on the Gimps project. He is now eligible for a $3,000 award.

When asked about mathematicians fascination with such mammoth numbers, Caldwell said: They are exciting to those of us who are interested in them. Its like asking why do you climb a mountain. He compares prime numbers to diamonds, with small ones finding uses in encryption and other applications, but large ones being more like showpieces. Thats what were talking about here: its a museum piece as opposed to something that industry would use, he said.

Curtis Cooper, a professor of mathematics at the University of Central Missouri, found the previous record-holding Mersenne prime in 2016, the fourth prime he has helped to find through the Gimps project in 20 years. He said he was a little sad at having lost the record so soon, but added: Im really happy for the whole organisation and the guy who found it. Hed been searching for 14 years, so hes worked as hard as I have.

Discovering new primes, which are things you can touch, its the realisation of my love for mathematics. Thats the appeal for me, he said.

Read more: https://www.theguardian.com/science/2018/jan/04/largest-prime-number-discovered-with-more-than-23m-digits

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15 Paradoxes That Will Make Your Head Explode

“I know one thing,” Socrates famously said. “That I know nothing.”

It’s a crucial insight from one of the founders of Western philosophy: You should question everything you think you know. 

Indeed, the closer you look, the more you’ll start to recognize paradoxes all around you.

Read on to see our favorite Catch-22s from Wikipedia’s epic list of more than 200 types of paradoxes.

To go anywhere, you must go halfway first, and then you must go half of the remaining distance, and half of the remaining distance, and so forth to infinity: Thus, motion is impossible.

The dichotomy paradox has been attributed to ancient Greek philosopher Zeno, and it was supposedly created as a proof that the universe is singular and that change, including motion, is impossible (as posited by Zeno’s teacher, Parmenides).

People have intuitively rejected this paradox for years.

From a mathematical perspective, the solution — formalized in the 19th century — is to accept that one-half plus one-quarter plus one-eighth plus one-sixteenth and so on … adds up to one. This is similar to saying that 0.999… equals 1.

But this theoretical solution doesn’t actually answer how an object can reach its destination. The solution to that question is more complex and still murky, relying on 20th-century theories about matter, time, and space not being infinitely divisible.

Mark Ramsay/Flickr CC BY 2.0

In any instant, a moving object is indistinguishable from a nonmoving object: Thus motion is impossible.

This is called the arrow paradox, and it’s another of Zeno’s arguments against motion. The issue here is that in a single instant of time, zero seconds pass, and so zero motion happens. Zeno argued that if time were made up of instants, the fact that motion doesn’t happen in any particular instant would mean motion doesn’t happen.

As with the dichotomy paradox, the arrow paradox actually hints at modern understandings of quantum mechanics. In his book “Reflections on Relativity,” Kevin Brown notes that in the context of special relativity, an object in motion is different from an object at rest. Relativity requires that objects moving at different speeds will appear different to outside observers and will themselves have different perceptions of the world around them.

If you restored a ship by replacing each of its wooden parts, would it remain the same ship?

Another classic from ancient Greece, the Ship of Theseus paradox gets at the contradictions of identity. It was famously described by Plutarch:

The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.

Can an omnipotent being create a rock too heavy for itself to lift?

While we’re at it, how can evil exist if God is omnipotent? And how can free will exist if God is omniscient?

These are a few of the many paradoxes that exist when you try to apply logic to definitions of God.

Some people might cite these paradoxes as reasons not to believe in a supreme being; however, others would say they are inconsequential or invalid.

Woodcut for “die Bibel in Bildern”, 1960 Wikimedia Commons

There’s an infinitely long “horn” that has a finite volume but an infinite surface area.

Moving ahead to a problem posed in the 17th century, we’ve got one of many paradoxes related to infinity and geometry.

“Gabriel’s Horn” is formed by taking the curve y = 1/x and rotating it around the horizontal axis, as shown in the picture. Using techniques from calculus that make it possible to calculate areas and volumes of shapes constructed this way, it’s possible to see that the infinitely long horn actually has a finite volume equal to π, but an infinite surface area.

As stated in the MathWorld article on the horn, this means that the horn could hold a finite volume of paint but would require an infinite amount of paint to cover its entire surface.

A heterological word is one that does not describe itself. Does “heterological” describe itself?

Here is one of many self-referential paradoxes that kept modern mathematicians and logicians up at night.

An example of a heterological word is “verb,” which is not a verb (as opposed to “noun,” which is itself a noun). Another example is “long,” which is not a long word (as opposed to “short,” which is a short word).

So is “heterological” a heterological word? If it were a word that didn’t describe itself, then it would describe itself; but if it did describe itself, then it would not be a word that described itself.

This is related to Russell’s Paradox, which asked if the set of things that don’t contain themselves contained itself. By creating self-destructing sets like these, Bertrand Russell and others showed the importance of establishing careful rules when creating sets, which would lay the groundwork for 20th-century mathematics.

Pilots can get out of combat duty if they are psychologically unfit, but anyone who tries to get out of combat duty proves he is sane.

Catch-22,” a satirical World War II novel by Joseph Heller, named the situation where someone is in need of something that can only be had by not being in need of it — which is a kind of self-referential paradox.

Protagonist Yossarian is introduced to the paradox with regard to pilot evaluation but eventually sees paradoxical (and oppressive) rules everywhere he looks.

S. Alexis/Flickr CC BY-SA 2.0

There is something interesting about every number.

After all, 1 is the first nonzero natural number; 2 is the smallest prime number; 3 is the first odd prime number; 4 is the smallest composite number; etc. And when you finally reach a number that seems not to have anything interesting about it, then that number is interesting by virtue of being the first number that is not interesting.

The Interesting Number Paradox relies on an imprecise definition of “interesting,” making this a somewhat sillier version of some of the other paradoxes, like the heterological paradox, that rely on contradictory self-references.

Quantum-computing researcher Nathaniel Johnston came up with a clever resolution of the paradox: Instead of relying on an intuitive notion of “interesting” as in the original paradox, he defined an interesting whole number as one appearing somewhere in the Online Encyclopedia of Integer Sequences, a collection of tens of thousands of mathematical sequences like the prime numbers, the Fibonacci numbers, or the Pythagorean triples.

Based on this definition, as of Johnston’s initial blog post in June 2009, the first uninteresting number — the smallest whole number that didn’t show up in any of the sequences — was 11,630. Since new sequences are added to the encyclopedia all the time, some of which include previously uninteresting numbers, as of Johnston’s most recent update in November 2013, the current smallest uninteresting number is 14,228.

In a bar, there is always at least one customer for whom it is true that if he is drinking, everyone is drinking.

Conditional statements in formal logic sometimes have counterintuitive interpretations, and the drinking paradox is a great example.

At first glance, the paradox suggests that one person is causing the rest of the bar to drink.

In fact, all it’s saying is that it would be impossible for everyone in the bar to be drinking unless every single customer were drinking. Therefore, there is at least one customer there (i.e., the last customer not drinking) who by drinking could make it so that everyone in the bar was drinking.

Employees Only/Facebook

A ball that can be cut into a finite number of pieces can be reassembled into two balls of the same size.

The Banach-Tarski paradox relies on a lot of the strange and counterintuitive properties of infinite sets and geometric rotations.

The pieces that the ball gets cut into are very strange-looking, and the paradox only works for an abstract, mathematical sphere: As nice as it would be to take an apple, cut it up, and reassemble the pieces so you have an extra apple for your friend, physical balls made of matter can’t be disassembled like a purely mathematical sphere.

A 100-gram potato is 99% water. If it dries to become 98% water, it will weigh only 50 grams.

Even when working with old-fashioned finite quantities, math can lead to strange results.

The key to the potato paradox is to closely look at the math behind the nonwater content of the potato. Since the potato is 99% water, the dry components are 1% of its mass. The potato starts at 100 grams, so that means that it contains 1 gram of dry material. When the dried-out potato is 98% water, that 1 gram of dry material now needs to account for 2% of the potato’s weight. One gram is 2% of 50 grams, so this must be the new weight of the potato.

If just 23 people are in a room, there’s a better-than-even chance at least two of them have the same birthday.

Another surprising math result, the birthday paradox comes from a careful analysis of the probabilities involved. If two people are in a room together, then there’s a 364/365 chance they do not have the same birthday (if we ignore leap years and assume that all birthdays are equally likely), since there are 364 days that are different from the first person’s birthday that can then be the second person’s birthday.

If there are three people in the room, then the probability that they all have different birthdays is 364/365 x 363/365: As above, once we know the first person’s birthday, there are 364 choices of a different birthday for the second person, and this leaves 363 choices for the third person’s birthday that are different from those two.

Continuing in this fashion, once you hit 23 people, the probability that all 23 have different birthdays drops below 50%, and so the probability that at least two have the same birthday is better than even.

Flickr / Les Roches International School of Hotel Management CC BY 2.0

Most people’s friends have more friends than they do.

This seems impossible but is true when you consider the math.

The friendship paradox is caused by how, in most social networks, most people have a few friends, while a handful of people have a large number of friends. Those social butterflies in the second group disproportionately show up as friends of people with smaller numbers of friends, and drag up the average number of friends-of-friends accordingly.

A physicist working on inventing the time machine is visited by an older version of himself. The older version gives him the plans for a time machine, and the younger version uses those plans to build the time machine, eventually going back in time as the older version of himself.

Time travel, if possible, could result in some extremely strange situations.

The bootstrap paradox is the opposite of the classic grandfather paradox: Rather than going back in time and preventing oneself from going back in time, some information or object is brought back in time, becoming a “younger” version of itself, and enabling itself later to travel back in time. One then has to ask: How did that information or object come into being in the first place?

The bootstrap paradox is common in science fiction and takes its name from a short story by Robert Heinlein

The Hubble eXtreme Deep Field (HXDF) taken in 2012/NASA Wikimedia Commons

If there’s nothing particularly unique about Earth, then there should be lots of alien civilizations in our galaxy. However, we’ve found no evidence of other intelligent life in the universe.

Finally, some see the silence of our universe as a paradox.

One of the underlying assumptions in astronomy is that Earth is a pretty common planet in a pretty common solar system in a pretty common galaxy, and that there is nothing cosmically unique about us. NASA’s Kepler satellite has found evidence that there are probably 11 billion Earth-like planets in our galaxy. Given this, life somewhat like us should have evolved somewhere not overly far away from us (at least on a cosmic scale).

But despite developing ever-more-powerful telescopes, we have had no evidence of technological civilizations anywhere else in the universe. Civilizations are noisy: Humanity broadcasts TV and radio signals that are unmistakably artificial. A civilization like ours should leave evidence that we would find.

Furthermore, a civilization that evolved millions of years ago (pretty recent from a cosmic perspective) would have had plenty of time to at least begin colonizing the galaxy, meaning there should be even more evidence of their existence. Indeed, given enough time, a colonizing civilization would be able to colonize the entire galaxy over the course of millions of years.

The physicist Enrico Fermi, for whom this paradox was named, simply asked, “Where are they?” in the middle of a lunchtime discussion with his colleagues. One resolution of the paradox challenges the above idea that Earth is common and posits instead that complex life is extremely rare in the universe. Another posits that technological civilizations inevitably wipe themselves out through nuclear war or ecological devastation.

A more optimistic solution is the idea that the aliens are intentionally hiding themselves from us until we become more socially and technologically mature. Yet another idea is that alien technology is so advanced that we wouldn’t even be able to recognize it.

Read more: http://www.iflscience.com/editors-blog/15-paradoxes-that-will-make-your-head-explode/

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The ‘imminent mini ice age’ myth is back, and it’s still wrong | Dana Nuccitelli

Dana Nuccitellil: We cant accurately predict solar activity, and a quiet solar cycle would have a small impact on Earths climate anyway

Roughly every two years were treated to headlines repeating the myth that Earth is headed for an imminent mini ice age. It happened in 2013, 2015, and again just recently at the tail end of 2017.

This time around, the myth appears to have been sparked by a Sky News interview with Northumbria University mathematics professor Valentina Zharkova. The story was quickly echoed by the Daily Mail, International Business Times, Sputnik News, Metro, Tru News, and others. Zharkova was also behind the mini ice age stories in 2015, based on her research predicting that the sun will soon enter a quiet phase.

The most important takeaway point is that the scientific research is clear were one to occur, a grand solar minimum would temporarily reduce global temperatures by less than 0.3C, while humans are already causing 0.2C warming per decade.

The
The global mean temperature difference is shown for the time period 1900 to 2100 for the IPCC A2 emissions scenario. The red line shows predicted temperature change for the current level of solar activity, the blue line shows predicted temperature change for solar activity at the much lower level of the Maunder Minimum, and the black line shows observed temperatures through 2010. Illustration: Adapted from Feulner & Rahmstorf (2010) in Geophysical Research Letters by SkepticalScience.com

So the sun could only offset at most 15 years worth of human-caused global warming, and once its quiet phase ended, the sun would then help accelerate global warming once again.

The mini ice age misnomer

The myth ultimately stems from a period climate scientists have coined The Little Ice Age (LIA). This was a modestly cool period running from about the year 1300 to 1850. It was particularly cold in the UK, where the River Thames sometimes froze over, and frost fairs were held.

A team led by University of Reading physicist and solar expert Mike Lockwood wrote a paper reviewing the science behind frost fairs, sunspots, and the LIA. It included the figure below showing northern hemisphere temperatures along with sunspot number and the level of volcanic particles in the atmosphere over the past millennium:

Sunspot
Sunspot number, northern hemisphere temperatures, and volcanic aerosol optical depth (AOD) around the time of the Little Ice Age. Illustration: Lockwood et al. (2017), News & Reviews in Astronomy & Geophysics

During full blown ice ages, temperatures have generally been 48C colder than in modern times. As this figure shows, during the LIA, temperatures were at most only about 0.5C cooler than the early 20th century. Thus, Lockwood calls the Little Ice Age a total misnomer. As the authors put it:

Compared to the changes in the proper ice ages, the so-called Little Ice Age (LIA) is a very short-lived and puny climate and social perturbation.

For comparison, temperatures have risen by a full 1C over the past 120 years, and 0.7C over just the past 40 years.

The minimal solar minima influence on the climate

The Maunder Minimum was a period of quiet solar activity between about 1645 and 1715. Its often referred to interchangeably with Little Ice Age, but the latter lasted centuries longer. In fact, three separate solar minima occurred during the LIA, which also included periods of relatively higher solar activity. Other factors like volcanic eruptions and human activities also contributed to the cool temperatures. In fact, a 2017 paper led by the University of Readings Mathew Owens concluded:

Climate model simulations suggest multiple factors, particularly volcanic activity, were crucial for causing the cooler temperatures in the northern hemisphere during the LIA. A reduction in total solar irradiance likely contributed to the LIA at a level comparable to changing land use [by humans].

Simulated
Simulated northern hemisphere temperature changes resulting from individual climate factors, as compared to the observed changes in the top panel. The bottom panel shows a simulation with no changes to climatological factors, to illustrate the level of natural variability in the climate. Illustration: Owens et al. (2017), Journal of Space Weather and Space Climate

Several studies have investigated the potential climate impact of a future grand solar minimum. In every case, they have concluded that such a quiet solar period would cause less than 0.3C cooling, which as previously noted, would temporarily offset no more than a decade and a halfs worth of human-caused global warming. These model-based estimates are consistent with the amount of cooling that occurred during the solar minima in the LIA.

Is another grand solar minimum imminent?

Although it would have a relatively small impact on the climate, its still an interesting question to ask whether were headed for another quiet solar period. Zharkova thinks so. Her team created a model that tries to predict solar activity, and suggests another solar minimum will occur from 2020 to 2055. However, other solar scientists have criticized the model as being too simple, created based on just 35 years of data, and failing to accurately reproduce past solar activity.

Ilya Usoskin, head of the Oulu Cosmic Ray Station and Vice-Director of the ReSoLVE Center of Excellence in Research, published a critique of Zharkovas solar model making those points. Most importantly, the model fails in reproducing past known solar activity because Zharkovas team treats the sun as a simple, predictable system like a pendulum. In reality, the sun has more random and unpredictable (in scientific terms, stochastic) behavior:

For example, a perfect pendulum if you saw a few cycles of the pendulum, you can predict its behavior. However, solar activity is known to be non-stationary process, which principally cannot be predicted (the prediction horizon for solar activity is known to be 10-15 years). Deterministic prediction cannot be made because of the essential stochastic component.

Just imagine a very turbulent flow of water in a river rapid, and you throw a small wooden stick into water and trace it. Then you do it second time and third time … each time the stick will end up in very different positions after the same time period. Its movement is unpredictable because of the turbulent stochastic component. This is exactly the situation with solar activity.

Lockwood agrees that we dont yet have a proven predictive theory of solar behavior. He has published research examining the range of possible solar evolutions based on past periods when the Sun was in a similar state to today, but as he puts it, that is the best that I think we can do at the present time!

Solar physicist Paul Charbonneau at the University of Montreal also concurred with Usoskin. He told me that while scientists are working to simulate solar activity, including using simplified models like Zharkovas,

on the standards of contemporary dynamo models theirs is extremely simple in fact borderlining simplistic … To extrapolate such a model outside its calibration window, you need an extra, very strong hypothesis: that the physical systems underlying the magnetic field generation retain their coherence (Phase, amplitude, etc.). As my colleague Ilya Usoskin has already explained, this is very unlikely to be the case in the case of the solar activity cycle.

Why wont this myth die?

Zharkova believes her solar model is correct, but at best it can only try to predict when the next quiet solar period will occur. Its influence on Earths climate is outside her expertise, and the peer-reviewed research is clear that it would be a minimal impact.

Zharkova disagrees I contacted her, and she told me that she believes a grand solar minimum would have a much bigger cooling effect. However, she also referenced long-debunked myths about global warming on Mars and Jupiter, and made a comment about the preachers of global warming. Shes clearly passionate about her research, and has the credibility that comes with publishing peer-reviewed studies on solar activity. Perhaps these factors motivate journalists to write these frequent mini ice age stories.

But Zharkovas climate science beliefs are irrelevant. While she has created a model predicting an imminent period of quiet solar activity, other scientists have identified serious flaws in the model, and in any case, research has shown that another solar minimum would only have a small and temporary impact on Earths climate.

Read more: https://www.theguardian.com/environment/climate-consensus-97-per-cent/2018/jan/09/the-imminent-mini-ice-age-myth-is-back-and-its-still-wrong

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19-Year-Old Student Hides Spy Camera In His Clothing To Take Secret Street Photos In The 1890s

Carl Størmer (1872-1957) enjoyed a hobby that was very, very unusual at the time. He walked around Oslo, Norway in the 1890s with his spy camera and secretly took everyday pictures of people. The subjects in Størmer’s pictures appear in their natural state. It extremely differs from the grave and strict posing trends that dominated in photography during those years.

Carl got his C.P. Stirn Concealed Vest Spy Camera in 1893 when he was studying mathematics at the Royal Frederick University (now, University of Oslo). “It was a round flat canister hidden under the vest with the lens sticking out through a buttonhole,” he told St. Hallvard Journal from in 1942. “Under my clothes I had a string down through a hole in my trouser pocket, and when I pulled the string the camera took a photo.”

Norway’s first paparazzi usually photographed people at the exact time they were greeting him on the street. “I strolled down Carl Johan, found me a victim, greeted, got a gentle smile and pulled. Six images at a time and then I went home to switch [the] plate.” In total, Størmer took a total of about 500 secret images.

His candid photos aside, Størmer was also fascinated with science. He was a mathematician and physicist, known both for his work in number theory and studying the Northern Lights (Aurora Borealis).

http://www.boredpanda.com/spy-camera-secret-street-photography-carl-stormer-norway/

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