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  • richardmitnick 11:04 am on September 11, 2019 Permalink | Reply
    Tags: "The answer to life, 32 is unsolvable., 65-year-old problem about 42, and everything", Andrew Booker of Bristol University, Are there three cubes whose sum is 42?, Diophantine Equation x3+y3+z3=k, Drew Sutherland, , , The universe   

    From MIT News: “The answer to life, the universe, and everything” 

    MIT News

    From MIT News

    September 10, 2019
    Sandi Miller | Department of Mathematics

    MIT mathematician Andrew “Drew” Sutherland solved a 65-year-old problem about 42. Image: Department of Mathematics

    This plot by Andrew Sutherland depicts the computation times for each of the 400,000-plus jobs that his team ran on Charity Engine’s compute grid. Each job was assigned a range of the parameter d = [x+y|, which must be a divisor of |z^3-42| for any integer solution to x^3 + y^3 + z^3 = 42. Each dot in the plot represents 25 jobs plotted according to their median runtime, with purple dots representing “smooth” values of d (those with no large prime divisors), and blue dots representing non-smooth values of d — the algorithm handles these two cases differently.
    Image: Andrew Sutherland

    Mathematics researcher Drew Sutherland helps solve decades-old sum-of-three-cubes puzzle, with help from “The Hitchhiker’s Guide to the Galaxy.”

    A team led by Andrew Sutherland of MIT and Andrew Booker of Bristol University has solved the final piece of a famous 65-year old math puzzle with an answer for the most elusive number of all: 42.

    The number 42 is especially significant to fans of science fiction novelist Douglas Adams’ “The Hitchhiker’s Guide to the Galaxy,” because that number is the answer given by a supercomputer to “the Ultimate Question of Life, the Universe, and Everything.”

    Booker also wanted to know the answer to 42. That is, are there three cubes whose sum is 42?

    This sum of three cubes puzzle, first set in 1954 at the University of Cambridge and known as the Diophantine Equation x3+y3+z3=k, challenged mathematicians to find solutions for numbers 1-100. With smaller numbers, this type of equation is easier to solve: for example, 29 could be written as 33 + 13 + 13, while 32 is unsolvable. All were eventually solved, or proved unsolvable, using various techniques and supercomputers, except for two numbers: 33 and 42.

    Booker devised an ingenious algorithm and spent weeks on his university’s supercomputer when he recently came up with a solution for 33. But when he turned to solve for 42, Booker found that the computing needed was an order of magnitude higher and might be beyond his supercomputer’s capability. Booker says he received many offers of help to find the answer, but instead he turned to his friend Andrew “Drew” Sutherland, a principal research scientist in the Department of Mathematics. “He’s a world’s expert at this sort of thing,” Booker says.

    Sutherland, whose specialty includes massively parallel computations, broke the record in 2017 for the largest Compute Engine cluster, with 580,000 cores on Preemptible Virtual Machines, the largest known high-performance computing cluster to run in the public cloud.

    Like other computational number theorists who work in arithmetic geometry, he was aware of the “sum of three cubes” problem. And the two had worked together before, helping to build the L-functions and Modular Forms Database(LMFDB), an online atlas of mathematical objects related to what is known as the Langlands Program. “I was thrilled when Andy asked me to join him on this project,” says Sutherland.

    Booker and Sutherland discussed the algorithmic strategy to be used in the search for a solution to 42. As Booker found with his solution to 33, they knew they didn’t have to resort to trying all of the possibilities for x, y, and z.

    “There is a single integer parameter, d, that determines a relatively small set of possibilities for x, y, and z such that the absolute value of z is below a chosen search bound B,” says Sutherland. “One then enumerates values for d and checks each of the possible x, y, z associated to d. In the attempt to crack 33, the search bound B was 1016, but this B turned out to be too small to crack 42; we instead used B = 1017 (1017 is 100 million billion).

    Otherwise, the main difference between the search for 33 and the search for 42 would be the size of the search and the computer platform used. Thanks to a generous offer from UK-based Charity Engine, Booker and Sutherland were able to tap into the computing power from over 400,000 volunteers’ home PCs, all around the world, each of which was assigned a range of values for d. The computation on each PC runs in the background so the owner can still use their PC for other tasks.

    Sutherland is also a fan of Douglas Adams, so the project was irresistible.

    The method of using Charity Engine is similar to part of the plot surrounding the number 42 in the “Hitchhiker” novel: After Deep Thought’s answer of 42 proves unsatisfying to the scientists, who don’t know the question it is meant to answer, the supercomputer decides to compute the Ultimate Question by building a supercomputer powered by Earth … in other words, employing a worldwide massively parallel computation platform.

    “This is another reason I really liked running this computation on Charity Engine — we actually did use a planetary-scale computer to settle a longstanding open question whose answer is 42.”

    They ran a number of computations at a lower capacity to test both their code and the Charity Engine network. They then used a number of optimizations and adaptations to make the code better suited for a massively distributed computation, compared to a computation run on a single supercomputer, says Sutherland.

    Why couldn’t Bristol’s supercomputer solve this problem?

    “Well, any computer *can* solve the problem, provided you are willing to wait long enough, but with roughly half a million PCs working on the problem in parallel (each with multiple cores), we were able to complete the computation much more quickly than we could have using the Bristol machine (or any of the machines here at MIT),” says Sutherland.

    Using the Charity Engine network is also more energy-efficient. “For the most part, we are using computational resources that would otherwise go to waste,” says Sutherland. “When you’re sitting at your computer reading an email or working on a spreadsheet, you are using only a tiny fraction of the CPU resource available, and the Charity Engine application, which is based on the Berkeley Open Infrastructure for Network Computing (BOINC), takes advantage of this. As a result, the carbon footprint of this computation — related to the electricity our computations caused the PCs in the network to use above and beyond what they would have used, in any case — is lower than it would have been if we had used a supercomputer.”

    Sutherland and Booker ran the computations over several months, but the final successful run was completed in just a few weeks. When the email from Charity Engine arrived, it provided the first solution to x3+y3+z3=42:

    42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3

    “When I heard the news, it was definitely a fist-pump moment,” says Sutherland. “With these large-scale computations you pour a lot of time and energy into optimizing the implementation, tweaking the parameters, and then testing and retesting the code over weeks and months, never really knowing if all the effort is going to pay off, so it is extremely satisfying when it does.”

    Booker and Sutherland say there are 10 more numbers, from 101-1000, left to be solved, with the next number being 114.

    But both are more interested in a simpler but computationally more challenging puzzle: whether there are more answers for the sum of three cubes for 3.

    “There are four very easy solutions that were known to the mathematician Louis J. Mordell, who famously wrote in 1953, ‘I do not know anything about the integer solutions of x3 + y3 + z3 = 3 beyond the existence of the four triples (1, 1, 1), (4, 4, -5), (4, -5, 4), (-5, 4, 4); and it must be very difficult indeed to find out anything about any other solutions.’ This quote motivated a lot of the interest in the sum of three cubes problem, and the case k=3 in particular. While it is conjectured that there should be infinitely many solutions, despite more than 65 years of searching we know only the easy solutions that were already known to Mordell. It would be very exciting to find another solution for k=3.”

    See the full article here .

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  • richardmitnick 9:43 am on May 6, 2016 Permalink | Reply
    Tags: , , , , The universe   

    From Ethan Siegel: “Top 10 Facts About The Big Bang Theory” 

    Starts with a Bang

    May 5, 2016
    Ethan Siegel

    Inflationary Universe. NASA/WMAP
    Inflationary Universe. NASA/WMAP


    Universe map Sloan Digital Sky Survey (SDSS) 2dF Galaxy Redshift Survey
    Universe map Sloan Digital Sky Survey (SDSS) 2dF Galaxy Redshift Survey

    SDSS Telescope at Apache Point, NM, USA
    SDSS Telescope at Apache Point, NM, USA

    If you ask a scientist where the Universe got its start, “the Big Bang” is the answer you’re most likely to get. Our Universe full of stars, galaxies and a cosmic web of large-scale structure, all separated by the vastness of empty space between them, wasn’t born that way and didn’t exist that way forever. Instead, the Universe came to be this way because it expanded and cooled from a hot, dense, uniform, matter-and-radiation-filled state with no galaxies, stars, or even atoms present at the start. Everything that exists in its current form today didn’t exist 13.8 billion years ago, and all of this was figured out during the past 100 years. But even with all this, there are a whole slew of facts most people — even many scientists — don’t quite get about it. Here are our top 10 facts about the Big Bang!

    Images credit: New York Times, 10 November 1919 (L); Illustrated London News, 22 November 1919 (R)

    1.) Einstein first dismissed it outright when it was presented to him as a possibility. Einstein’s general theory of relativity was a revolutionary theory of gravity, proposed in 1915, as a successor to Newton’s theory. It predicted the orbital motion of Mercury to an accuracy Newton’s theory couldn’t, it predicted the bending of starlight by mass confirmed in 1919…

    Radio galaxies gravitationally lensed by a very large foreground galaxy cluster Hubble
    Radio galaxies gravitationally lensed by a very large foreground galaxy cluster Hubble

    …and it predicted the existence of gravitational waves, just confirmed a few months ago.

    Caltech/MIT Advanced aLigo detector in Livingston, Louisiana
    Caltech/MIT Advanced aLigo detector in Livingston, Louisiana

    Gravitational waves. Credit: MPI for Gravitational Physics/W.Benger-Zib
    Gravitational waves. Credit: MPI for Gravitational Physics/W.Benger-Zib

    But it also predicted that a Universe that was full of matter and static, or unchanging over time, would be unstable.

    Georges Lemaître at the Catholic University of Leuven, ca. 1933. Public domain image.

    When the Belgian priest and scientist Georges Lemaître, in 1927, put forth the idea that the spacetime fabric of the Universe could be very large and expanding, having emerged from a smaller, denser, more uniform state in the past, Einstein wrote back to him, “Vos calculs sont corrects, mais votre physique est abominable,” which means “ your calculations are correct, but your physics is abominable!”

    Image credit: Robert P. Kirshner, PNAS, via http://www.pnas.org/content/101/1/8/F3.expansion. The red box indicates the extent of Hubble’s original data.

    2.) Hubble’s discovery of the expanding Universe turned it into a serious idea. Although many scientists considered that the spiral nebulae in the sky were distant galaxies all on their own even before Einstein, it was Edwin Hubble’s work in the 1920s that showed this was not only true, but that the more distant a galaxy was, the faster it was receding away from us. This fact — Hubble’s Law, describing the expansion of the Universe — led to a very straightforward interpretation consistent with the Big Bang idea: if the Universe is expanding today, then it was smaller and denser in the past!

    3.) The idea had been around since 1922, but was widely dismissed for decades. Soviet Physicist Alexandr Friedmann came up with the theory for it in 1922, when it was criticized by Einstein. Lemaître’s 1927 work was also dismissed by Einstein, and even after Hubble’s work in 1929, the idea that the Universe was smaller, denser, and more uniform in the past was only a fringe idea. But Lemaître added in the idea that the redshift of galaxies could be explained by this expansion of space, and that there must have been an initial “moment of creation” at the beginning, which was known as either the “primeval atom” or the “cosmic egg” for decades.

    Into what is the universe expanding NASA Goddard
    Into what is the universe expanding NASA Goddard, Dana Berry.

    4.) The theory rose to true prominence in the 1940s when it made a startling set of predictions. George Gamow, an American scientist who became enamored of Lemaître’s ideas, realized that if the Universe was expanding today, then the wavelength of the light in it was increasing over time, and therefore the Universe was cooling. If it’s cooling today, then it must have been hotter in the past. Extrapolating backwards, he recognized that there once was a time period where it was too hot for neutral atoms to form, and then a period before that where it was too hot for even atomic nuclei to forms. Therefore, as the Universe expanded and cooled, it must have formed the light elements and then neutral atoms for the first time, resulting in the existence of a “primeval fireball,” or a cosmic background of cold radiation just a few degrees above absolute zero.

    Fred Hoyle presenting a radio series, The Nature of the Universe, in 1950. Image credit: BBC.

    5.) The name “Big Bang” came about from the theory’s most fervent detractor, Fred Hoyle. A theory making a different set of predictions — the Steady-State Theory of the Universe — was actually the leading theory of the Universe in the 1940s, 1950s and into the 1960s, as the claim that the vast majority of the atoms came from stars that died and not this early, hot dense state was borne out by nuclear physics. Hoyle, speaking to the BBC, coined the term in a 1949 radio interview, saying, “One [idea] was that the Universe started its life a finite time ago in a single huge explosion, and that the present expansion is a relic of the violence of this explosion. This big bang idea seemed to me to be unsatisfactory even before detailed examination showed that it leads to serious difficulties.”

    Big Ear, Arno Penzias and Robert Wilson, AT&T, Holmdel, NJ USA
    Big Ear, Arno Penzias and Robert Wilson, AT&T, Holmdel, NJ USA

    6.) The 1964 discovery of the leftover glow from the Big Bang was initially thought to be from bird poop. In 1964, scientists Arno Penzias and Bob Wilson, working at the Holmdel Horn Antenna at Bell Labs, discovered a uniform radio signal coming from everywhere in the sky at once. Not realizing it was the Big Bang’s leftover glow, they thought it was a problem with the antenna, and tried to calibrate this “noise” away. When that didn’t work, they went into the antenna and discovered nests of pigeons living in there! They cleaned the nests (and droppings) of the pigeons out of there, and yet the signal remained. The realization that it was the discovery of Gamow’s prediction vindicated the Big Bang model, entrenching it as the scientific origin of our Universe. It also makes Penzias and Wilson the only Nobel-winning scientists to clean up animal poop as part of their Nobel-worthy research.

    7.) The confirmation of the Big Bang gives us an explicit history for the formation of stars, galaxies, and rocky planets in the Universe. If the Universe started off hot, dense, expanding and uniform, then not only would we cool and form atomic nuclei and neutral atoms, but it would take time for gravitation to pull objects together into gravitationally collapsed structures. The first stars would take 50-to-100 million years to form; the first galaxies wouldn’t form for 150-250 million years; Milky Way-sized galaxies might take billions of years and the first rocky planets wouldn’t form until multiple generations of stars lived, burned through their fuel, and died in catastrophic supernovae explosions. It may not be a coincidence that we’re observing the Universe now, 13.8 billion years after the Big Bang; it might be that this is when the time is ripe for life on rocky worlds to emerge!

    Cosmic Microwave Background per ESA/Planck
    Cosmic Microwave Background per ESA/Planck


    8.) The fluctuations in the cosmic microwave background tell us how close-to-perfectly uniform the Universe was at the start of the Big Bang. The cosmic microwave background is just 2.725 K today, but the fluctuations shown above are only around ~100 microKelvin in magnitude. The fact that the leftover glow from the Big Bang has slight non-uniformities of a particular magnitude at that early time tells us that the Universe was uniform to 1-part-in-30,000, but the fluctuations are what give rise to all the structure — stars, galaxies, etc. — that we see in the Universe today.

    History of the universe, National Science Foundation, E Siegel
    National Science Foundation (NASA, JPL, Keck Foundation, Moore Foundation, related) — Funded BICEP2 Program; modifications by E. Siegel.

    9.) The Big Bang itself doesn’t necessarily mean the very beginning anymore. It’s tempting to extrapolate this hot, dense expanding state all the way back to a singularity, as Lemaître did some 89 years ago. But there’s a suite of observations — led by the fluctuations in the primeval fireball — that teach us there was a different state prior to that, where all the energy in the Universe was inherent to space itself, and that space expanded at an exponential rate. This period was known as cosmic inflation, and we’re still researching the details on that. Science progresses farther and farther back, but so far, there’s no end in sight.

    Image credit: NASA & ESA, of possible models of the expanding Universe.

    10.) And the way the Universe began doesn’t tell us the way it will end. Finally, the Big Bang tells us there was a race between gravity, trying to recollapse the expanding Universe, and the initial expansion, trying to drive everything apart. But the Big Bang on its own doesn’t tell us what the fate will be; that takes knowing what the entire Universe is made out of. With the existence of dark energy, discovered just 18 years ago, we’ve learned that not only will the expansion win, but that the most distant galaxies will continue to speed up in their recession from us. Our cold, lonely, empty fate is what we get in a dark energy Universe, but if the Universe were born with just a tiny bit more matter or radiation than what we have today, our fate could’ve been very different!

    See the full article here .

    Please help promote STEM in your local schools.

    STEM Icon

    Stem Education Coalition

    “Starts With A Bang! is a blog/video blog about cosmology, physics, astronomy, and anything else I find interesting enough to write about. I am a firm believer that the highest good in life is learning, and the greatest evil is willful ignorance. The goal of everything on this site is to help inform you about our world, how we came to be here, and to understand how it all works. As I write these pages for you, I hope to not only explain to you what we know, think, and believe, but how we know it, and why we draw the conclusions we do. It is my hope that you find this interesting, informative, and accessible,” says Ethan

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