Category Archives: numbers

On Fermat’s Last Theorem (Conjecture)

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Fermat, a French mathematician of the late 17th century, came up with a conjecture that baffled other mathematicians for three and half centuries until Andrew Wiles published a proof in the mid-nineties. Most of you are familiar from high school geometry with the Pythagorean theorem, that the sum of two integers squared may be equal to another integer squared: a2 + b2 = c2, but Fermat imagined a more general problem for integers where an + bn ≠ cn where n>2: Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. That last bit is the mystery—that the margin was too small for his proof. Many mathematicians believe he did not have a proof, but all the same, he did throw down the gauntlet by making the conjecture. He just wrote the conjecture, that an + bn = cn, is not possible. Wiles’ proof is so complex and convoluted, however, that you have to be a brilliant mathematician to even begin to understand his arguments. For as simple as the Pythagorean theorem looks, Fermat’s conjecture is inversely complex, and complex in ways that not even a great mathematician can dream. The conjecture looks simple, but the answer seems to be one of the most complex ever proved in the history of mathematics. The proof, almost as elusive as the Holy Grail, is unintelligible to the average lay person, and difficult for even the gifted. What kind of mind does it take to fathom the dark and profound reaches of Fermat’s conjecture? This conjecture, according to a French academy of math, has the dubious honor of having the highest number of incorrect proofs written about it. In other words, many mathematicians have tried to conquer the proof, but died ignominiously on the battlefield without having succeeded. That fact that Wiles did his work in secret suggests that even he thought the little problem might be paradoxically unsolvable—a no-win scenario, as it were, and a career-ending catastrophe. That there is, after all, a solution to Fermat’s last theorem is of little consolation to all of that failure. (Sorry mathematicians,formatting limitations don’t allow for the little raised numbers in the equations.)

On Fermat’s Last Theorem (Conjecture)

Posted on by

Fermat, a French mathematician of the late 17th century, came up with a conjecture that baffled other mathematicians for three and half centuries until Andrew Wiles published a proof in the mid-nineties. Most of you are familiar from high school geometry with the Pythagorean theorem, that the sum of two integers squared may be equal to another integer squared: a2 + b2 = c2, but Fermat imagined a more general problem for integers where an + bn ≠ cn where n>2: Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. That last bit is the mystery—that the margin was too small for his proof. Many mathematicians believe he did not have a proof, but all the same, he did throw down the gauntlet by making the conjecture. He just wrote the conjecture, that an + bn = cn, is not possible. Wiles’ proof is so complex and convoluted, however, that you have to be a brilliant mathematician to even begin to understand his arguments. For as simple as the Pythagorean theorem looks, Fermat’s conjecture is inversely complex, and complex in ways that not even a great mathematician can dream. The conjecture looks simple, but the answer seems to be one of the most complex ever proved in the history of mathematics. The proof, almost as elusive as the Holy Grail, is unintelligible to the average lay person, and difficult for even the gifted. What kind of mind does it take to fathom the dark and profound reaches of Fermat’s conjecture? This conjecture, according to a French academy of math, has the dubious honor of having the highest number of incorrect proofs written about it. In other words, many mathematicians have tried to conquer the proof, but died ignominiously on the battlefield without having succeeded. That fact that Wiles did his work in secret suggests that even he thought the little problem might be paradoxically unsolvable—a no-win scenario, as it were, and a career-ending catastrophe. That there is, after all, a solution to Fermat’s last theorem is of little consolation to all of that failure. (Sorry mathematicians,formatting limitations don’t allow for the little raised numbers in the equations.)

On recycling

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Most people probably get recycling, although for some the trash is just everything you want to throw away, and that means everything–every container, can, bottle, box, and paper goes into the trash, no discrimination. I think those days have long since past, however, when we can afford to just throw it all away. Recycling is about knowing that the planet’s resources are finite, and that we must reuse and recycle almost everything that we can. Recycling seems like an obvious response to the insanity of filling up an endless string of landfills with valuable resources. These are moral and ethical choices we are making that affect us now and will affect our children in the decades to come. Recyclable materials, whether they are metal, glass, plastic or paper, are valuable commodities that with a little effort can be turned in to new products. Yet, when driving around my own neighborhood on any given Friday morning when the trash cans have been wheeled out to the curb one can find lots of recyclable materials protruding from the gray bins whose contents will be going to the landfill. Perhaps I am wrong in assuming that the general public understands the finite nature of natural resources and how quickly we are using them up. No paper or cardboard should ever go into the general trash, yet one might spot a huge cardboard box roughly jammed into the top of the gray trash bins as if it were a corpse with the feet sticking out. Why would anyone throw away an aluminum can when the metal dealers will give you real money for the empties? Let’s say one is addicted to diet sodas: a real drinker might turn all of those empty cans into some real money by the end of the month, but lots of cans just go into the garbage, or tossed onto the side of the road, or left in the gutter for someone else to pick up. Our relationship with our garbage is an unhappy one, marked by dysfunction and bitterness. Wouldn’t it be better to just keep two bins at home? One for organic waste which has little value and should be disposed of, and one for all of the materials that can be recycled. The volume of the organic waste is very small, but all of the bottles, cans, boxes, and paper take up an enormous amount of space. In order to recycle one must be proactive and make an effort to distinguish between what is, or is not, trash. Of all the things we throw away, very little of it is actual trash, and if we could compost that without attracting vermin, we could reduce our solid wastes to almost nothing. The time will come when natural resources give out and we will have to dig up the dumps and land fills to “mine” all of the precious materials that lie smoldering away, buried years ago by a society that put no value in recycling. There are those people who do not look to the future, don’t really care about the planet, don’t understand how we are polluting our environment with landfills and such, don’t believe in global warming, and, in general, just don’t care about anything or anyone but themselves. The choice to not recycle, then, turns into a moral decision to just wallow in filth and ego and not care. I am not idealistic enough to believe that recycling will solve our problems regarding the use of natural resources, manufacturing, pollution, or greenhouse effects, but I do think there will come a time when it will be illegal to dump recyclables into the trash, a time which has already come to some communities across the USA. This is no longer a question of whether recycling is a good idea, it’s a question when we are going to take it seriously. Now? Or when we no longer have the choice?

On recycling

Posted on by

Most people probably get recycling, although for some the trash is just everything you want to throw away, and that means everything–every container, can, bottle, box, and paper goes into the trash, no discrimination. I think those days have long since past, however, when we can afford to just throw it all away. Recycling is about knowing that the planet’s resources are finite, and that we must reuse and recycle almost everything that we can. Recycling seems like an obvious response to the insanity of filling up an endless string of landfills with valuable resources. These are moral and ethical choices we are making that affect us now and will affect our children in the decades to come. Recyclable materials, whether they are metal, glass, plastic or paper, are valuable commodities that with a little effort can be turned in to new products. Yet, when driving around my own neighborhood on any given Friday morning when the trash cans have been wheeled out to the curb one can find lots of recyclable materials protruding from the gray bins whose contents will be going to the landfill. Perhaps I am wrong in assuming that the general public understands the finite nature of natural resources and how quickly we are using them up. No paper or cardboard should ever go into the general trash, yet one might spot a huge cardboard box roughly jammed into the top of the gray trash bins as if it were a corpse with the feet sticking out. Why would anyone throw away an aluminum can when the metal dealers will give you real money for the empties? Let’s say one is addicted to diet sodas: a real drinker might turn all of those empty cans into some real money by the end of the month, but lots of cans just go into the garbage, or tossed onto the side of the road, or left in the gutter for someone else to pick up. Our relationship with our garbage is an unhappy one, marked by dysfunction and bitterness. Wouldn’t it be better to just keep two bins at home? One for organic waste which has little value and should be disposed of, and one for all of the materials that can be recycled. The volume of the organic waste is very small, but all of the bottles, cans, boxes, and paper take up an enormous amount of space. In order to recycle one must be proactive and make an effort to distinguish between what is, or is not, trash. Of all the things we throw away, very little of it is actual trash, and if we could compost that without attracting vermin, we could reduce our solid wastes to almost nothing. The time will come when natural resources give out and we will have to dig up the dumps and land fills to “mine” all of the precious materials that lie smoldering away, buried years ago by a society that put no value in recycling. There are those people who do not look to the future, don’t really care about the planet, don’t understand how we are polluting our environment with landfills and such, don’t believe in global warming, and, in general, just don’t care about anything or anyone but themselves. The choice to not recycle, then, turns into a moral decision to just wallow in filth and ego and not care. I am not idealistic enough to believe that recycling will solve our problems regarding the use of natural resources, manufacturing, pollution, or greenhouse effects, but I do think there will come a time when it will be illegal to dump recyclables into the trash, a time which has already come to some communities across the USA. This is no longer a question of whether recycling is a good idea, it’s a question when we are going to take it seriously. Now? Or when we no longer have the choice?