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Documentary: Fermat's Last Theorem

Fermat's Last Theorem is one of the most famous math problems in history, as it remained unsolved for well over 300 years -- and Fermat himself, before his suicide, had written a margin note claiming he had a solution. Fellow _floss blogger Casey Johnston described FLT like so last year:

In 1637, Pierre de Fermat scribbled a note in the margin of his copy of the book Arithmetica. He wrote (conjectured, in math terms) that for an integer n greater that two, the equation an + bn = cn had no whole number solutions. He wrote a proof for the special case n = 4, and claimed to have a simple, “marvellous” proof that would make this statement true for all integers. However, Fermat was fairly secretive about his mathematic endeavors, and no one discovered his conjecture until his death in 1665. No trace was found of the proof Fermat claimed to have for all numbers, and so the race to prove his conjecture was on. For the next 330 years, many great mathematicians, such as Euler, Legendre, and Hilbert, stood and fell at the foot of what came to be known as Fermat’s Last Theorem. Some mathematicians were able to prove the theorem for more special cases, such as n = 3, 5, 10, and 14. Proving special cases gave a false sense of satisfaction; the theorem had to be proved for all numbers. Mathematicians began to doubt that there were sufficient techniques in existence to prove theorem. Eventually, in 1984, a mathematician named Gerhard Frey noted the similarity between the theorem and a geometrical identity, called an elliptical curve. Taking into account this new relationship, another mathematician, Andrew Wiles, set to work on the proof in secrecy in 1986. Nine years later, in 1995, with help from a former student Richard Taylor, Wiles successfully published a paper proving Fermat’s Last Theorem, using a recent concept called the Taniyama-Shimura conjecture. 358 years later, Fermat’s Last Theorem had finally been laid to rest.

In 1996, Simon Singh and John Lynch made a 45-minute documentary about Wiles and his FLT solution. It's a surprisingly engaging story, with twists and turns and math and problem-solving (also liberal use of Penguin Cafe Orchestra in the soundtrack). Wiles comes across as a genuine mathematician -- he was obsessed with FLT since boyhood, and his joy at its solution is clearly visible in this film. It's all available on Google Video, embedded below. Have a look if you're even vaguely interested in math! (Note: the math discussed in the film is aimed at a mainstream audience; you don't have to be a math whiz to get the human drama of the story.)

(Via Kottke.org.)

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science
Here's What Actually Happens When You're Electrocuted
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Benjamin Franklin was a genius, but not so smart when it came to safely handling electricity, according to legend. As SciShow explains in its latest video, varying degrees of electric current passing through the body can result in burns, seizures, cessation of breathing, and even a stopped heart. Our skin is pretty good at resisting electric current, but its protective properties are diminished when it gets wet—so if Franklin actually conducted his famous kite-and-key experiment in the pouring rain, he was essentially flirting with death.

That's right, death: Had Franklin actually been electrocuted, he wouldn't have had only sparks radiating from his body and fried hair. The difference between experiencing an electric shock and an electrocution depends on the amount of current involved, the voltage (the difference in the electrical potential that's driving the current), and your body's resistance to the current. Once the line is crossed, the fallout isn't pretty, which you can thankfully learn about secondhand by watching the video below.

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Big Questions
Does Einstein's Theory of Relativity Imply That Interstellar Space Travel is Impossible?
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Does Einstein's theory of relativity imply that interstellar space travel is impossible?

Paul Mainwood:

The opposite. It makes interstellar travel possible—or at least possible within human lifetimes.

The reason is acceleration. Humans are fairly puny creatures, and we can’t stand much acceleration. Impose much more than 1 g of acceleration onto a human for an extended period of time, and we will experience all kinds of health problems. (Impose much more than 10 g and these health problems will include immediate unconsciousness and a rapid death.)

To travel anywhere significant, we need to accelerate up to your travel speed, and then decelerate again at the other end. If we’re limited to, say, 1.5 g for extended periods, then in a non-relativistic, Newtonian world, this gives us a major problem: Everyone’s going to die before we get there. The only way of getting the time down is to apply stronger accelerations, so we need to send robots, or at least something much tougher than we delicate bags of mostly water.

But relativity helps a lot. As soon as we get anywhere near the speed of light, then the local time on the spaceship dilates, and we can get to places in much less (spaceship) time than it would take in a Newtonian universe. (Or, looking at it from the point of view of someone on the spaceship: they will see the distances contract as they accelerate up to near light-speed—the effect is the same, they will get there quicker.)

Here’s a quick table I knocked together on the assumption that we can’t accelerate any faster than 1.5 g. We accelerate up at that rate for half the journey, and then decelerate at the same rate in the second half to stop just beside wherever we are visiting.

You can see that to get to destinations much beyond 50 light years away, we are receiving massive advantages from relativity. And beyond 1000 light years, it’s only thanks to relativistic effects that we’re getting there within a human lifetime.

Indeed, if we continue the table, we’ll find that we can get across the entire visible universe (47 billion light-years or so) within a human lifetime (28 years or so) by exploiting relativistic effects.

So, by using relativity, it seems we can get anywhere we like!

Well ... not quite.

Two problems.

First, the effect is only available to the travelers. The Earth times will be much much longer. (Rough rule to obtain the Earth-time for a return journey [is to] double the number of light years in the table and add 0.25 to get the time in years). So if they return, they will find many thousand years have elapsed on earth: their families will live and die without them. So, even we did send explorers, we on Earth would never find out what they had discovered. Though perhaps for some explorers, even this would be a positive: “Take a trip to Betelgeuse! For only an 18 year round-trip, you get an interstellar adventure and a bonus: time-travel to 1300 years in the Earth’s future!”

Second, a more immediate and practical problem: The amount of energy it takes to accelerate something up to the relativistic speeds we are using here is—quite literally—astronomical. Taking the journey to the Crab Nebula as an example, we’d need to provide about 7 x 1020 J of kinetic energy per kilogram of spaceship to get up to the top speed we’re using.

That is a lot. But it’s available: the Sun puts out 3X1026 W, so in theory, you’d only need a few seconds of Solar output (plus a Dyson Sphere) to collect enough energy to get a reasonably sized ship up to that speed. This also assumes you can transfer this energy to the ship without increasing its mass: e.g., via a laser anchored to a large planet or star; if our ship needs to carry its chemical or matter/anti-matter fuel and accelerate that too, then you run into the “tyranny of the rocket equation” and we’re lost. Many orders of magnitude more fuel will be needed.

But I’m just going to airily treat all that as an engineering issue (albeit one far beyond anything we can attack with currently imaginable technology). Assuming we can get our spaceships up to those speeds, we can see how relativity helps interstellar travel. Counter-intuitive, but true.

This post originally appeared on Quora. Click here to view.

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