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Great Thinkers, Great Theorems

Great Thinkers, Great Theorems

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Great Thinkers, Great Theorems

Course No. 1471
Professor William Dunham, Ph.D.
Muhlenberg College
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4.7 out of 5
52 Reviews
94% of reviewers would recommend this series
Course No. 1471
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Course Overview

Mathematics is filled with beautiful theorems that are as breathtaking as the most celebrated works of art, literature, or music. They are the Mona Lisas, Hamlets, and Fifth Symphonys of the field—landmark achievements that repay endless study and that are the work of geniuses as fascinating as Leonardo, Shakespeare, and Beethoven. Here is a sample:

  • Pythagorean theorem: Although he didn't discover the Pythagorean theorem about a remarkable property of right triangles, the Greek mathematician Euclid devised an ingenious proof that is a mathematical masterpiece. Plus, it's beautiful to look at!
  • Area of a circle: The formula for the area of a circle, A = π r2, was deduced in a marvelous chain of reasoning by the Greek thinker Archimedes. His argument relied on the clever tactic of proof by contradiction not once, but twice.
  • Basel problem: The Swiss mathematician Leonhard Euler won his reputation in the early 1700s by evaluating an infinite series that had stumped the best mathematical minds for a generation. The solution was delightfully simple; the path to it, bewilderingly complex.
  • Larger infinities: In the late 1800s, the German mathematician Georg Cantor blazed the trail into the "transfinite" by proving that some infinite sets are bigger than others, thereby opening a strange new realm of mathematics.

You can savor these results and many more in Great Thinkers, Great Theorems, 24 half-hour lectures that conduct you through more than 3,000 years of beautiful mathematics, telling the story of the growth of the field through a carefully chosen selection of its most awe-inspiring theorems.

Approaching great theorems the way an art course approaches great works of art, the course opens your mind to new levels of math appreciation. And it requires no more than a grasp of high school mathematics, although it will delight mathematicians of all abilities.

Your guide on this lavishly illustrated tour, which features detailed graphics walking you through every step of every proof, is Professor William Dunham of Muhlenberg College, an award-winning teacher who has developed an artist's eye for conveying the essence of a mathematical idea. Through his enthusiasm for brilliant strategies, novel tactics, and other hallmarks of great theorems, you learn how mathematicians think and what they mean by "beauty" in their work. As added enrichment, the course guidebook has supplementary questions and problems that allow you to go deeper into the ideas behind the theorems.

An Innovative Approach to Mathematics

Professor Dunham has been taking this innovative approach to mathematics for over a quarter-century—in the classroom and in his popular books. With Great Thinkers, Great Theorems you get to watch him bring this subject to life in stimulating lectures that combine history, biography, and, above all, theorems, presented as a series of intellectual adventures that have built mathematics into the powerful tool of analysis and understanding that it is today.

In the arts, a great masterpiece can transform a genre; think of Claude Monet's 1872 canvas Impression, Sunrise, which gave the name to the Impressionist movement and revolutionized painting. The same is true in mathematics, with the difference that the revolution is permanent. Once a theorem has been established, it is true forever; it never goes out of style. Therefore the great theorems of the past are as fresh and impressive today as on the day they were first proved.

What Makes a Theorem Great?

A theorem is a mathematical proposition backed by a rigorous chain of reasoning, called a proof, that shows it is indisputably true. As for greatness, Professor Dunham believes the defining qualities of a great theorem are elegance and surprise, exemplified by these cases:

  • Elegance: Euclid has a beautifully simple way of showing that any finite collection of prime numbers can't be complete—that there is always at least one prime number left out, proving that the prime numbers are infinite. Dr. Dunham calls this one of the greatest proofs in all of mathematics.
  • Surprise: Another Greek, Heron, devised a formula for triangular area that is so odd that it looks like it must be wrong. "It's my favorite result from geometry just because it's so implausible," says Dr. Dunham, who shows how, 16 centuries later, Isaac Newton used algebra in an equally surprising route to the same result.

Great Thinkers, Great Theorems includes many lectures that are devoted to a single theorem. In these, Professor Dunham breaks the proof into manageable pieces so that you can follow it in detail. When you get to the Q.E.D.—the initials traditionally ending a proof, signaling quod erat demonstrandum (Latin for "that which was to be demonstrated")—you can step back and take in the masterpiece as a whole, just as you would with a painting in a museum.

In other lectures, you focus on the biographies of the mathematicians behind these masterpieces—geniuses who led eventful, eccentric, and sometimes tragic lives. For example:

  • Cardano: Perhaps the most bizarre mathematician who ever lived, the 16th-century Italian Gerolamo Cardano was a gambler, astrologer, papal physician, convicted heretic, and the first to publish the solution of cubic and quartic algebraic equations, which he did after a no-holds-barred competition with rival mathematicians.
  • Newton and Leibniz: The battle over who invented calculus, the most important mathematical discovery since ancient times, pitted Isaac Newton—mathematician, astronomer, alchemist—against Gottfried Wilhelm Leibniz— mathematician, philosopher, diplomat. Each believed the other was trying to steal the credit.
  • Euler: The most inspirational story in the history of mathematics belongs to Leonhard Euler, whose astonishing output barely slowed down after he went blind in 1771. Like Beethoven, who composed some of his greatest music after going deaf, Euler was able to practice his art entirely in his head.
  • Cantor: While Vincent van Gogh was painting pioneering works of modern art in France in the late 1800s, Georg Cantor was laying the foundations for modern mathematics next door in Germany. Unappreciated at first, the two rebels even looked alike, and both suffered debilitating bouts of depression.

Describing a common reaction to the theorems produced by these great thinkers, Professor Dunham says his students often want to know where the breakthrough ideas came from: How did the mathematicians do it? The question defies analysis, he says. "It's like asking: ‘Why did Shakespeare put the balcony scene in Romeo and Juliet? What made him think of it?' Well, he was Shakespeare. This is what genius looks like!" And by watching the lectures in Great Thinkers, Great Theorems, you will see what equivalent genius looks like in mathematics.

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24 lectures
 |  30 minutes each
  • 1
    Theorems as Masterpieces
    Certain theorems stand out as great masterpieces of mathematics that can be appreciated as great works of art. After hearing Professor Dunham explain this approach, discover the two ways of proving a theorem: direct proof and indirect proof. Also, meet some of the great thinkers whose ideas you will be studying. x
  • 2
    Mathematics before Euclid
    Investigate three non-Greek civilizations that had robust traditions in mathematics. Then encounter a pair of Greek mathematicians who predated Euclid, but who left very deep footprints: Thales and Pythagoras—the latter renowned for the theorem that bears his name. x
  • 3
    The Greatest Mathematics Book of All
    Begin your exploration of the work widely considered the greatest mathematical text of all time: Euclid's Elements. Discover why these 13 succinct books have been so influential for so long as you delve into the ground-laying definitions, postulates, common notions, and theorems from book I. x
  • 4
    Euclid's Elements—Triangles and Polygons
    Continuing your journey through Euclid, work your way toward his most famous result: his proof of the Pythagorean theorem—a demonstration of remarkable visual and intellectual beauty. Also, cover some of the techniques from book IV for constructing regular polygons. x
  • 5
    Number Theory in Euclid
    In addition to being a geometer, Euclid was a pioneering number theorist, a subject he took up in books VII, VIII, and IX of the Elements. Focus on his proof that there are infinitely many prime numbers, which Professor Dunham considers one of the greatest proofs in all of mathematics. x
  • 6
    The Life and Works of Archimedes
    Even more distinguished than Euclid was Archimedes, whose brilliant ideas took centuries to fully absorb. Probe the life and famous death of this absent-minded thinker, who once ran unclothed through the streets, shouting "Eureka!" ("I have found it!") on solving a problem in his bath. x
  • 7
    Archimedes' Determination of Circular Area
    See Archimedes in action by following his solution to the problem of determining circular area—a question that seems trivial today but only because he solved it so simply and decisively. His unusual strategy relied on a pair of indirect proofs. x
  • 8
    Heron's Formula for Triangular Area
    Heron of Alexandria (also called Hero) is known as the inventor of a proto-steam engine many centuries before the Industrial Revolution. Discover that he was also a great mathematician who devised a curious method for determining the area of a triangle from the lengths of its three sides. x
  • 9
    Al-Khwarizmi and Islamic Mathematics
    With the decline of classical civilization in the West, the focus of mathematical activity shifted to the Islamic world. Investigate the proofs of the mathematician whose name gives us our term "algorithm": al-Khwarizmi. His great book on equation solving also led to the term "algebra." x
  • 10
    A Horatio Algebra Story
    Visit the ruthless world of 16th-century Italian universities, where mathematicians kept their discoveries to themselves so they could win public competitions against their rivals. Meet one of the most colorful of these figures: Gerolamo Cardano, who solved several key problems. In secret, of course. x
  • 11
    To the Cubic and Beyond
    Trace Cardano's path to his greatest triumph: the solution to the cubic equation, widely considered impossible at the time. His protégé, Ludovico Ferrari, then solved the quartic equation. Norwegian mathematician Niels Abel later showed that no general solutions are possible for fifth- or higher-degree equations. x
  • 12
    The Heroic Century
    The 17th century saw the pace of mathematical innovations accelerate, not least in the introduction of more streamlined notation. Survey the revolutionary thinkers of this period, including John Napier, Henry Briggs, René Descartes, Blaise Pascal, and Pierre de Fermat, whose famous "last theorem" would not be proved until 1995. x
  • 13
    The Legacy of Newton
    Explore the eventful life of Isaac Newton, one of the greatest geniuses of all time. Obsessive in his search for answers to questions from optics to alchemy to theology, he made his biggest mark in mathematics and science, inventing calculus and discovering the law of universal gravitation. x
  • 14
    Newton's Infinite Series
    Start with the binomial expansion, then turn to Newton's innovation of using fractional and negative exponents to calculate roots—an example of his creative use of infinite series. Also see how infinite series allowed Newton to approximate sine values with extraordinary accuracy. x
  • 15
    Newton's Proof of Heron's Formula
    Return to Heron's ancient formula for determining the area of a triangle to consider Newton's proof using algebraic techniques—an approach he also applied to other geometry problems. The steps are circuitous, but the result bears Newton's stamp of genius. x
  • 16
    The Legacy of Leibniz
    Probe the career of Newton's great rival, Gottfried Wilhelm Leibniz, who came relatively late to mathematics, plunging in during a diplomatic assignment to Paris. In short order, he discovered the "Leibniz series" to represent π, and within a few years he invented calculus independently of Newton. x
  • 17
    The Bernoullis and the Calculus Wars
    Follow the bitter dispute between Newton and Leibniz over priority in the development of calculus. Also encounter the Swiss brothers Jakob and Johann Bernoulli, enthusiastic supporters of Leibniz. Their fierce sibling rivalry extended to their competition to outdo each other in mathematical discoveries. x
  • 18
    Euler, the Master
    Meet history's most prolific mathematician, Leonhard Euler, who went blind in his sixties but kept turning out brilliant papers. A sampling of his achievements: the number e, crucial in calculus; Euler's identity, responsible for the most beautiful theorem ever; Euler's polyhedral formula; and Euler's path. x
  • 19
    Euler's Extraordinary Sum
    Euler won his spurs as a great mathematician by finding the value of a converging infinite series that had stumped the Bernoulli brothers and everyone else who tried it. Pursue Euler's analysis through the twists and turns that led to a brilliantly simple answer. x
  • 20
    Euler and the Partitioning of Numbers
    Investigate Euler's contribution to number theory by first warming up with the concept of amicable numbers—a truly rare breed of integers until Euler vastly increased the supply. Then move on to Euler's daring proof of a partitioning property of whole numbers. x
  • 21
    Gauss—the Prince of Mathematicians
    Dubbed the Prince of Mathematicians by the end of his career, Carl Friedrich Gauss was already making major contributions by his teen years. Survey his many achievements in mathematics and other fields, focusing on his proof that a regular 17-sided polygon can be constructed with compass and straightedge alone. x
  • 22
    The 19th Century—Rigor and Liberation
    Delve into some of the important trends of 19th-century mathematics: a quest for rigor in securing the foundations of calculus; the liberation from the physical sciences, embodied by non-Euclidean geometry; and the first significant steps toward opening the field to women. x
  • 23
    Cantor and the Infinite
    Another turning point of 19th-century mathematics was an increasing level of abstraction, notably in the approach to the infinite taken by Georg Cantor. Explore the paradoxes of the "completed" infinite, and how Cantor resolved this mystery with transfinite numbers, exemplified by the transfinite cardinal aleph-naught. x
  • 24
    Beyond the Infinite
    See how it's possible to build an infinite set that's bigger than the set of all whole numbers, which is itself infinite. Conclude the course with Cantor's theorem that the transcendental numbers greatly outnumber the seemingly more abundant algebraic numbers—a final example of the elegance, economy, and surprise of a mathematical masterpiece. x

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Your professor

William Dunham

About Your Professor

William Dunham, Ph.D.
Muhlenberg College
Dr. William Dunham is the Truman Koehler Professor of Mathematics at Muhlenberg College in Allentown, Pennsylvania. He earned his undergraduate degree from the University of Pittsburgh and his M.S. and Ph.D. in Mathematics from The Ohio State University. Before his current appointment at Muhlenberg, Dr. Dunham taught at Hanover College in Indiana, receiving teaching awards from both institutions as well as the Award for...
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Great Thinkers, Great Theorems is rated 4.7 out of 5 by 52.
Rated 5 out of 5 by from A great treat. I received this as a Christmas gift. What a joy it has been to listen to Dr. Dunham as he traces the history of great mathematicians and their math throughout history. This parallels, in a fascinating way, the study of great artists and their art, or, great writers and their literature. You don't have to be a math-wiz to thoroughly enjoy this excellent series. I haven't taken a math class in over 35 years, but this series is delightful. The anecdotes he gives make it especially appealing.
Date published: 2017-03-27
Rated 3 out of 5 by from Not what was promised. I am very sorry to say that I will be returning this course, as it was not at all what it was advertised to be. The course description states that: "Mathematics is filled with beautiful theorems that are as breathtaking as the most celebrated works of art, literature, or music. They are the Mona Lisas, Hamlets, and Fifth Symphonys of the field—landmark achievements that repay endless study and that are the work of geniuses as fascinating as Leonardo, Shakespeare, and Beethoven." Undoubtedly, this is true, so I jumped into these lectures with great eagerness to appreciate, contemplate, and bask in the beauty of the mathematical equivalents to the creations by DaVinci, Shakespeare, and Beethoven. I expected to see the great equations presented as great works of art, in the way I recall Jacob Bronowski presenting the Pythagorean Theorem in the wonderful BBC documentary "The Ascent of Man", or perhaps to be presented with the mathematical wonders underlying the beauty of harmony (or color, for that matter), or perhaps to spend some time contemplating the mathematical beauty animating the golden ratio and its multifarious expressions both in nature and in the great masterpieces of human art. I wanted to moved, in short, in the same way I am moved when I stand before the Mona Lisa (or, better yet, when I am contemplating Raphael's School of Athens or Michelangelo's divine creation on the ceiling of the Sistine Chapel), or the way that I am left in awe after reading Hamlet, or the way I have goosebumps all over my body (and, sometimes, tears) after listening to Beethoven's Fifth. I wasn't. What I got instead was an extremely pleasant professor, who knows a lot about his subject, giving in-depth (and interesting) biological sketches of his subjects, and then spending some time discussing some of their important contributions to mathematics. Don't get my wrong. I enjoyed both the biographical and the mathematical discussions, but this is not the course I bought. There was a certain amount of pleasure gained in following the derivation of an important theorem, but rarely did I feel the professor to be presenting these theorems as being "as breathtaking as the most celebrated works of art, literature, or music." Never, in short, did the professor ever explain the beauty behind the most beautiful equation in the entire history of mathematics, Euler's identity, an equation that many mathematicians (and even some non-mathematicians) even frame and hang on their walls as a work of art! This, to me, was inexcusable, especially in light of the description of lecture 18). Perhaps this was the fault of the marketing department rather than the professor--it is hard to say. Had the course been marketed as a course in which there would be a discussion of some of the more interesting and important mathematical theorems, combined with brief biographical sketches of the famous mathematicians who propounded them, then this would have been at least a 4- (and maybe a 5-) star course. As I have said, there was substance to these lectures, and the professor did have a pleasant teaching style. However, as a course that marketed itself as being concerned with the artistic beauty of mathematical ideas, it came up woefully short. The merits of the course (good substance and a pleasing style from the professor) overcame some of the shortcomings of the course, resulting in (to my, anyway) a 3-star rating.
Date published: 2017-03-21
Rated 5 out of 5 by from Very enthusiastic Lecturer. They Love the topic The lecturer gave great introductions and clear examples. I had the video version, which I think you need to understand this course. I like the calm presentation, with good spirited comments on the topics. They clearly loved math and all the theorems revealed through the centuries.
Date published: 2017-03-14
Rated 5 out of 5 by from Looking at Mathematics in History Before I begin, just a note to give the reader some perspective. I taught secondary mathematics in both public and private schools for 30 years, so I have my basics down pat, fully aware that seven out of four people struggle with fractions and ratios. Those people will be glad to know Great Thinkers, Great Theorems is not really a math course, rather a history of math class, centered on some of math's greatest names. It brings many of the concepts we all studied in high school into a better perspective. The course is highly informative and thought-provoking. I have always believed interest in mathematics would be boosted if students really knew where it came from, and how awesome it really is, considering none of these folks had computers or internet. My only criticism of the course is slight. The professor is not as in awe of Descartes as I am. This, “I think, therefore, I am,” Frenchman opened the door to analytic geometry, leading to calculus by adding a vertical number line to the familiar horizontal one, allowing us to plot any point on a plane. While he is honored as the “Father of graph paper,” this simple move was pure genius. Anyway, if you’ve ever wondered where all this cursed math comes from, you will discover many of the answers in this course an intellectual delight. As for Professor Dunham, I found him most amusing, as did my wife and daughter when they watched these lectures with me. His style is a bit jerky, but his knowledge and dry sense of humor add to the enjoyment of this class. It’s probably not for everyone, but if you enjoy being challenged and amazed by some of the great thinkers of the past, you will delight in these lectures as your eyes are opened to what can only be described as pure genius. Enjoy.
Date published: 2017-03-13
Rated 5 out of 5 by from A first-rate lecture series from every standpoint. Of the 75 Great Courses I have purchased over the years, this is one of the top five.
Date published: 2017-01-15
Rated 5 out of 5 by from Fine Humor and Deep Knowledge of Maths As my professional formation is Law (I am Chief Justice in Rio de Janeiro, Brazil) I purchased this Course after watched another one of the same subject (indicated below). I always loved Mathematics during my school years but had to abandoned it when reached the University. Professor William Dunham is phenomenal. He is elegant, has a nice voice and a fine humor to present the 24 lectures of the Course. He explained us the historical context and some insights of the lives of famous mathematicians, beginning with Euclides. He talks about their main books and main formulas and how they got them. Each lecture is easy to follow and not demand any previous mathematical understanding. Highly recommended for anyone who wants to know how the great theorems were formulated and are important till today.
Date published: 2017-01-15
Rated 5 out of 5 by from Recommended for all levels Whether you have a college level math background or just basic high school background, you will enjoy the history, biographical material and the basic proofs in this very entertaining course.
Date published: 2017-01-14
Rated 4 out of 5 by from Overall, a very good exposition to the development Overall, a very good exposition to the development of mathematics from its beginning to current times. I was really interested in the choice of the top mathematicians. The teacher includes a picture of each, with a touch of humor regarding the 19th century – we finally have photographs. The booklet includes a very helpful list of references with the teacher’s personal review of each book. I really appreciated the major contribution from classical Greece – the cradle of Western civilization. As opposed to the same ideas from India and China, the Greek masters had the originality of working out general solutions instead of a unique solution based on a specific length and also establishing proofs based on logic to demonstrate why a formula is correct. All this in spite of having no algebra and an archaic numbering system unsuitable for calculations. Lectures 2 to 5 provide a very good summary of Euclid’s Elements masterpiece of geometry which remains valid to this day. Less technically-minded viewers might be put off by the demonstrations but there really is more to the course than what appears to be dry math. In lecture 7 Mr. Dunham conveys enthusiasm while explaining how Archimedes calculated the area of a circle at a time where algebra did not exist and how closely he was nibbling at the edge of mathematical limits and calculus. His brilliant problem-solving approach yielded the correct result about two millennia before Newton developed calculus – once learned, calculus enables anyone to mechanically follow rules to arrive directly at the formula without the need for Archimedes’ great intuition and genius. It is fascinating to learn how Arab mathematicians adopted, preserved and expanded upon the classical Greece masterworks, after the Greek influence on mathematics came to an end as Rome was overrun by the Huns, the Plato academy was shut down, the library of accumulated knowledge in Alexandria was ravaged by fire and Europe entered the sterile Middle Age. The work by Arab mathematicians, augmented by their adoption of the Hindu decimal numbering system – complete with the zero – then appeared in Italy and Spain during the Renaissance and was translated back into Latin. Followed by Gutenberg’s invention of the printing press a few centuries later ensuring that mathematical masterworks would be preserved and distributed widely. Lectures 11 and 15 go beyond casual listening and might turn off some – Mr. Dunham even labels lecture 15 as “high-school math with a vengeance”, a long proof based on simple math. This can be skipped without harm and you can always come back later. However, 15 is not that difficult and you do not have to raise your hand and interrupt the teacher – just hit rewind – you’ll get the hang of it and an appreciation on how mathematicians prove formulas. The way a simple calculation is carried out in advance and saved for re-use later is particularly effective. During the development of the main topic you suddenly arrive at an incomplete step which – by no mere coincidence – is just the calculation that was done ahead of time, greatly easing the understanding of the entire proof. This should server as an example for teachers to follow. Lecture 16 on Leibniz starts off with an amazing demonstration of the power of insight, showing an extremely simple way to obtain the exact value of an infinite series. Lectures 17 - 20 convey great enthusiasm over Euler. A reviewer is always biased. There can be no other way since the course reviewer is presenting a personal appreciation of the course. Here, the biases are this reviewer’s training as an engineer in applied science as well as Grossman’s Thermodynamics and Wolfson’s Physics monumental works as the absolute 5-star course reference, Dunham’s course has to be in the 4-star set. Not because of any fault of its own, it just does not reach the bar set really high by Grossman and Wolfson. Tough to be the star in a roomful of aces. On the negative side – as suggestions for improvements (or a follow-up course EXPLAINING the great mathematical tools used in applied sciences: what the Fourier and Laplace transforms are, etc,?) I found Lecture 1 to be as s…l…o…w as molasses in winter. Motivating a course is good, but this was overdone. Glad I decided to keep at it instead of dismissing the entire course outright, in spite of a less-than-stellar first impression. Lecture 20 was definitely not my favorite. Number theory is apparently not something I care for very much. It would have been in my opinion much more interesting to expand upon the use of the e number – try asking an engineer how he could do without it. Or showing how “logarithms doubled an astronomer’s life.” Also missing is an explanation of what problem caused Newton to develop calculus, his insight in doing it. I also think Gauss’s work with Weber on magnetism deserved more detailed treatment instead of a cursory mention – the first two equations of electromagnetism are named “Gauss’s’ law of electricity” and “Gauss’s law of magnetism”. In lecture 22 on the entry of women into mathematics there is no mention of Emmy Noether who spectacularly linked the constancy of physics throughout space with electrical charge conservation and the constancy of physics throughout time with energy conservation. Probably not “liberated” enough from physics – my personal bias again. This lecture also celebrates the “liberation” of mathematics from science thus elevating mathematics to a higher level – this felt a bit like renouncing its parents. One flaw of lecture 11 which deals with solving a cubic equation. On the plus side, a geometric cube is decomposed as a series of smaller cubes, each one representing a term of the equation. Unfortunately, the teacher just shows a representation in two dimensions (with a perspective for the third dimension) – it would have been so much clearer to actually cut wooden blocks labelled with the proper sizes that could be assembled as the whole block. Due to this flaw, I find I have to take Mr. Dunham’s word for it because there are too many of them and I found it difficult to visualize the components. But this is no big deal, just a missed opportunity.
Date published: 2016-12-29
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