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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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Women Suffer Worse Migraines Than Men. Now Scientists Think They Know Why
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Migraines are one of medicine's most frustrating mysteries, both causes and treatments. Now researchers believe they've solved one part of the puzzle: a protein affected by fluctuating estrogen levels may explain why more women suffer from migraines than men.

Migraines are the third most common illness in the world, affecting more than 1 in 10 people. Some 75 percent of sufferers are women, who also experience them more frequently and more intensely, and don't respond as well to drug treatments as men do.

At this year's Experimental Biology meeting in San Diego, researcher Emily Galloway presented new findings on the connection between the protein NHE1 and the development of migraine headaches. NHE1 regulates the transfer of protons and sodium ions across cell membranes, including the membranes that separate incoming blood flow from the brain.

When NHE1 levels are low or the molecule isn't working as it's supposed to, migraine-level head pain can ensue. And because irregular NHE1 disrupts the flow of protons and sodium ions to the brain, medications like pain killers have trouble crossing the blood-brain barrier as well. This may explain why the condition is so hard to treat.

When the researchers analyzed NHE1 levels in the brains of male and female lab rats, the researchers found them to be four times higher in the males than in the females. Additionally, when estrogen levels were highest in the female specimens, NHE1 levels in the blood vessels of their brains were at their lowest.

Previous research had implicated fluctuating estrogen levels in migraines, but the mechanism behind it has remained elusive. The new finding could change the way migraines are studied and treated in the future, which is especially important considering that most migraine studies have focused on male animal subjects.

"Conducting research on the molecular mechanisms behind migraine is the first step in creating more targeted drugs to treat this condition, for men and women," Galloway said in a press statement. "Knowledge gained from this work could lead to relief for millions of those who suffer from migraines and identify individuals who may have better responses to specific therapies."

The new research is part of a broader effort to build a molecular map of the relationship between sex hormones and NHE1 expression. The next step is testing drugs that regulate these hormones to see how they affect NHE1 levels in the brain.

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History
A Founder of Earth Day Looks Back on How It Began
Vivien Killilea/Getty Images for Caruso Affiliated

On the very first Earth Day in 1970, Denis Hayes stood on a stage in Central Park, stunned by the number of people who'd come to honor the planet. Now in his 70s, Hayes remembers it was like looking at the ocean—“you couldn’t see where the sea of people ended.” Crowd estimates reached more than a million people.

For Hayes, who is now board chair of the international Earth Day Network, it was the culmination of a year’s worth of work. As an urban ecology graduate student at Harvard University, he’d volunteered to help organize a small initiative by Wisconsin senator Gaylord Nelson. Nelson was horrified by the 1969 oil spill in Santa Barbara, California, and wanted to raise awareness about environmental issues by holding teaching events similar to those being held by civil rights and anti-war activists.

Senator Nelson saw a growing disconnect between the concept of progress and the idea of American well-being, Hayes tells Mental Floss. “There was a sense that America was prosperous and getting better, but at the same time, the air in the country was similar to the air today in China, Mexico City, or New Delhi," Hayes says. "Rivers were catching on fire. Lakes were unswimmable.”

Nelson's plan for these environmental teach-ins was for speakers to educate college students about environmental issues. But he had no one to organize them. So Hayes, Nelson’s sole volunteer, took control on a national level, organizing teach-ins at Harvard first and then across the U.S. Initially, the response was tepid at best. “Rather rapidly it became clear that this wasn’t a hot issue at colleges and universities in 1969,” Hayes says. “We had a war raging, and civil rights were getting very emotional after the Nixon election.”

Still, both Hayes and Nelson noticed an influx of mail to the senator's office from women with young families worried about the environment. So instead of focusing on colleges, the two decided to take a different tactic, creating events with community-based organizations across the country, Hayes says. They also decided that rather than a series of teach-ins, they'd hold a single, nationwide teach-in on the same day. They called it Earth Day, and set a date: April 22.

Hayes now had a team of young adults working for the cause, and he himself had dropped out of school to tackle it full time. Long before social media, the project began to spread virally. “It just resonated,” he says. Women and smaller environmental-advocacy groups really hooked onto the idea, and word spread by mouth and by information passing between members of the groups.

Courtesy of Denis Hayes

With the cooperation and participation of grassroots groups and volunteers across the country, and a few lawmakers who supported the initiative, Hayes’ efforts culminated in the event on April 22, 1970.

Hayes started the day in Washington, D.C., where he and the staff were based. There was a rally and protest on the National Mall, though by that point Hayes had flown to New York, where Mayor John Lindsay provided a stage in Central Park. Parts of Fifth Avenue were shut down for the events, which included Earth-oriented celebrations, protests, and speeches by celebrities. Some of those attending the event even attacked nearby cars for causing pollution. After the rally, Hayes flew to Chicago for a smaller event.

“We had a sense that it was going to be big, but when the day actually dawned, the crowds were so much bigger than anyone had experienced before,” Hayes said. The event drew grassroots activists working on a variety of issues—Agent Orange, lead paint in poor urban neighborhoods, saving the whales—and fostered a sense of unity among them.

“There were people worrying about these [environmental] issues before Earth Day, but they didn’t think they had anything in common with one another," Hayes says. "We took all those individual strands and wove them together into the fabric of modern environmentalism.”

Hayes and his team spent the summer getting tear-gassed at protests against the American invasion of Cambodia, which President Nixon authorized just six days after Earth Day. But by fall, the team refocused on environmental issues—and elections. They targeted a “dirty dozen” members of Congress up for re-election who had terrible environmental records, and campaigned for candidates who championed environmental causes to run against them. They defeated seven out of 12.

“It was a very poorly funded but high-energy campaign,” Hayes says. “That sent the message to Congress that it wasn’t just a bunch of people out frolicking in the sunshine planting daisies and picking up litter. This actually had political chops.”

The early '70s became a golden age for environmental issues; momentum from the Earth Day movement spawned the creation of the Clean Air Act, the Clean Water Act, the Safe Drinking Water Act, the Endangered Species Act, the Marine Mammal Protection Act, the Environmental Education Act (which was initially passed in 1970 and revived in 1990), and the Environmental Protection Agency.

“We completely changed the framework within which America does business, more than any other period in history with the possible exception of the New Deal,” Hayes says. “But our little revolution was brought entirely from the grassroots up.”

In 1990, Hayes was at it again. He organized the first international Earth Day, with about 200 million participants across more than 140 countries. Since then it’s become a global phenomenon.

Despite its popularity, though, we still have a long way to go, even if the improvements Hayes fought for have made these issues feel more remote. Hayes noted that everything they were fighting in the '70s was something tangible—something you could see, taste, smell, or touch. Climate change can seem much less real—and harder to combat—to the average person who isn’t yet faced with its effects.

Hayes also notes that people have become more skeptical of science. “Historically, that has not been a problem in the United States. But today science is under attack.”

He warns, “This [anti-science sentiment] is something that could impoverish the next 50 generations and create really long-term devastation—that harms not only American health, but also American business, American labor, and American prospects.”

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