Fermat Theorem

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation has no integer solutions for 2" width="29" border="0" height="14"> and .

The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, 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" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

As a result of Fermat's marginal note, the proposition that the Diophantine equation

(1)

where , , , and are integers, has no nonzero solutions for 2" width="29" border="0" height="14"> has come to be known as Fermat's Last Theorem. It was called a "theorem" on the strength of Fermat's statement, despite the fact that no other mathematician was able to prove it for hundreds of years.

Note that the restriction 2" width="29" border="0" height="14"> is obviously necessary since there are a number of elementary formulas for generating an infinite number of Pythagorean triples satisfying the equation for ,

(2)

A first attempt to solve the equation can be made by attempting to factor the equation, giving

(3)

Since the product is an exact power,

(4)

Solving for and gives

(5)

which give

(6)

However, since solutions to these equations in rational numbers are no easier to find than solutions to the original equation, this approach unfortunately does not provide any additional insight.

If an odd prime divides , then the reduction

(7)

can be made, so redefining the arguments gives

(8)

If no odd prime divides , then is a power of 2, so and, in this case, equations (7) and (8) work with 4 in place of . Since the case was proved by Fermat to have no solutions, it is sufficient to prove Fermat's last theorem by considering odd prime powers only.

Similarly, is sufficient to prove Fermat's last theorem by considering only relatively prime , , and , since each term in equation (1) can then be divided by , where is the greatest common divisor.

The so-called "first case" of the theorem is for exponents which are relatively prime to , , and () and was considered by Wieferich. Sophie Germain proved the first case of Fermat's Last Theorem for any odd prime when is also a prime. Legendre subsequently proved that if is a prime such that , , , , or is also a prime, then the first case of Fermat's Last Theorem holds for . This established Fermat's Last Theorem for . In 1849, Kummer proved it for all regular primes and composite numbers of which they are factors (Vandiver 1929, Ball and Coxeter 1987).

The "second case" of Fermat's last theorem is " divides exactly one of , , . Note that is ruled out by , , being relatively prime, and that if divides two of , , , then it also divides the third, by equation (8).

Kummer's attack led to the theory of ideals, and Vandiver developed Vandiver's criteria for deciding if a given irregular prime satisfies the theorem. In 1852, Genocchi proved that the first case is true for if is not an irregular pair. In 1858, Kummer showed that the first case is true if either or is an irregular pair, which was subsequently extended to include and by Mirimanoff (1909). Vandiver (1920ab) pointed out gaps and errors in Kummer's memoir which, in his view, invalidate Kummer's proof of Fermat's Last Theorem for the irregular primes 37, 59, and 67, although he claims Mirimanoff's proof of FLT for exponent 37 is still valid.

Wieferich (1909) proved that if the equation is solved in integers relatively prime to an odd prime , then

(9)

(Ball and Coxeter 1987). Such numbers are called Wieferich primes. Mirimanoff (1909) subsequently showed that

(10)

must also hold for solutions relatively prime to an odd prime , which excludes the first two Wieferich primes 1093 and 3511. In 1914, Vandiver showed

(11)

and Frobenius extended this to

(12)

It has also been shown that if were a prime of the form , then

(13)

which raised the smallest possible in the "first case" to by 1941 (Rosser 1941). Granville and Monagan (1988) showed if there exists a prime satisfying Fermat's Last Theorem, then

(14)

for , 7, 11, ..., 71. This establishes that the first case is true for all prime exponents up to (Vardi 1991).

The "second case" of Fermat's Last Theorem (for ) proved harder than the first case.

Euler proved the general case of the theorem for , Fermat , Dirichlet and Lagrange . In 1832, Dirichlet established the case . The case was proved by Lamé (1839; Wells 1986, p. 70), using the identity

(15)

Although some errors were present in this proof, these were subsequently fixed by Lebesgue in 1840. Much additional progress was made over the next 150 years, but no completely general result had been obtained. Buoyed by false confidence after his proof that pi is transcendental, the mathematician Lindemann proceeded to publish several proofs of Fermat's Last Theorem, all of them invalid (Bell 1937, pp. 464-465). A prize of German marks, known as the Wolfskehl Prize, was also offered for the first valid proof (Ball and Coxeter 1987, p. 72; Barner 1997; Hoffman 1998, pp. 193-194 and 199).

A recent false alarm for a general proof was raised by Y. Miyaoka (Cipra 1988) whose proof, however, turned out to be flawed. Other attempted proofs among both professional and amateur mathematicians are discussed by vos Savant (1993), although vos Savant erroneously claims that work on the problem by Wiles (discussed below) is invalid. By the time 1993 rolled around, the general case of Fermat's Last Theorem had been shown to be true for all exponents up to (Cipra 1993). However, given that a proof of Fermat's Last Theorem requires truth for all exponents, proof for any finite number of exponents does not constitute any significant progress towards a proof of the general theorem (although the fact that no counterexamples were found for this many cases is highly suggestive).

In 1993, a bombshell was dropped. In that year, the general theorem was partially proven by Andrew Wiles (Cipra 1993, Stewart 1993) by proving the semistable case of the Taniyama-Shimura conjecture. Unfortunately, several holes were discovered in the proof shortly thereafter when Wiles' approach via the Taniyama-Shimura conjecture became hung up on properties of the Selmer group using a tool called an Euler system. However, the difficulty was circumvented by Wiles and R. Taylor in late 1994 (Cipra 1994, 1995) and published in Taylor and Wiles (1995) and Wiles (1995). Wiles' proof succeeds by (1) replacing elliptic curves with Galois representations, (2) reducing the problem to a class number formula, (3) proving that formula, and (4) tying up loose ends that arise because the formalisms fail in the simplest degenerate cases (Cipra 1995).

The proof of Fermat's Last Theorem marks the end of a mathematical era. Since virtually all of the tools which were eventually brought to bear on the problem had yet to be invented in the time of Fermat, it is interesting to speculate about whether he actually was in possession of an elementary proof of the theorem. Judging by the tenacity with which the problem resisted attack for so long, Fermat's alleged proof seems likely to have been illusionary. This conclusion is further supported by the fact that Fermat searched for proofs for the cases and , which would have been superfluous had he actually been in possession of a general proof.

In the episode of the television program The Simpsons, the equation appeared at one point in the background. Expansion reveals that only the first 9 decimal digits match (Rogers 2005). The episode The Wizard of Evergreen Terrace mentions , which matches not only in the first 10 decimal places but also the easy-to-check last place (Greenwald).

SEE ALSO: abc Conjecture, Beal's Conjecture, Bogomolov-Miyaoka-Yau Inequality, Euler System, Fermat-Catalan Conjecture, Fermat's Theorem, Generalized Fermat Equation, Mordell Conjecture, Pythagorean Triple, Ribet's Theorem, Selmer Group, Sophie Germain Prime, Szpiro's Conjecture, Taniyama-Shimura Conjecture, Vojta's Conjecture, Waring Formula

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source

Quarks

Quarks are fundamental matter particles that are constituents of neutrons and protons and other hadrons. There are six different types of quarks. Each quark type is called a flavor.

Flavor		Mass (GeV/c2)	Electric Charge (e)
u	up	0.004	+2/3
d	down	0.008	-1/3
c	charm	1.5	+2/3
s	strange	0.15	-1/3
t	top	176	+2/3
b	bottom	4.7	-1/3

Quark Masses

Quarks only exist inside hadrons because they are confined by the strong (or color charge) force fields. Therefore, we cannot measure their mass by isolating them. Furthermore, the mass of a hadron gets contributions from quark kinetic energy and from potential energy due to strong interactions. For hadrons made of the light quark types, the quark mass is a small contribution to the total hadron mass. For example, compare the mass of a proton (0.938 GeV/c²) to the sum of the masses of two up quarks and one down quark (total of 0.02 GeV/c²).

So the question is, what do we mean by the mass of a quark and how do we measure it. The quantity we call quark mass is actually related to the m in F = ma (force = mass x acceleration). This equation tells us how an object will behave when a force is applied. The equations of particle physics include, for example, calculations of what happens to a quark when struck by a high energy photon. The parameter we call quark mass controls its acceleration when a force is applied. It is fixed to give the best match between theory and experiment both for the ratio of masses of various hadrons and for the behavior of quarks in high energy experiments. However, neither of these methods can precisely determine quark masses.

How Do We Know Quarks Are Real?

A question you might well ask! If we cannot separate them out, how do we know they are there? The answer is simply that all our calculations depend on their existence and give the right answers for the experiments.

For example, when we bounce electrons off of protons and neutrons, the pattern of scattering angles observed is characteristic of point-like spin-1/2 scatters. The relative rates for electron versus neutrino scattering is that predicted from the quark electric charges. The process of electron-positron annihilation to quark pairs gives similar characteristic predictions, all these are also confirmed experimentally. The accumulation of many such results, where experiments match predictions based on quarks, convinces us that quarks are real.

Superconductor History

Superconductors, materials that have no resistance to the flow of electricity, are one of the last great frontiers of scientific discovery. Not only have the limits of superconductivity not yet been reached, but the theories that explain superconductor behavior seem to be constantly under review. In 1911 superconductivity was first observed in mercury by Dutch physicist Heike Kamerlingh Onnes of Leiden University (shown above). When he cooled it to the temperature of liquid helium, 4 degrees Kelvin (-452F, -269C), its resistance suddenly disappeared. The Kelvin scale represents an "absolute" scale of temperature. Thus, it was necessary for Onnes to come within 4 degrees of the coldest temperature that is theoretically attainable to witness the phenomenon of superconductivity. Later, in 1913, he won a Nobel Prize in physics for his research in this area.
Walter Meissner

The next great milestone in understanding how matter behaves at extreme cold temperatures occurred in 1933. German researchers Walter Meissner (above) and Robert Ochsenfeld discovered that a superconducting material will repel a magnetic field (below graphic). A magnet moving by a conductor induces currents in the conductor. This is the principle on which the electric generator operates. But, in a superconductor the induced currents exactly mirror the field that would have otherwise penetrated the superconducting material - causing the magnet to be repulsed. This phenomenon is known as strong diamagnetism and is today often referred to as the "Meissner effect" (an eponym). The Meissner effect is so strong that a magnet can actually be levitated over a superconductive material.

In subsequent decades other superconducting metals, alloys and compounds were discovered. In 1941 niobium-nitride was found to superconduct at 16 K. In 1953 vanadium-silicon displayed superconductive properties at 17.5 K. And, in 1962 scientists at Westinghouse developed the first commercial superconducting wire, an alloy of niobium and titanium (NbTi). High-energy, particle-accelerator electromagnets made of copper-clad niobium-titanium were then developed in the 1960s at the Rutherford-Appleton Laboratory in the UK, and were first employed in a superconducting accelerator at the Fermilab Tevatron in the US in 1987.

The first widely-accepted theoretical understanding of superconductivity was advanced in 1957 by American physicists John Bardeen, Leon Cooper, and John Schrieffer (above). Their Theories of Superconductivity became know as the BCS theory - derived from the first letter of each man's last name - and won them a Nobel prize in 1972. The mathematically-complex BCS theory explained superconductivity at temperatures close to absolute zero for elements and simple alloys. However, at higher temperatures and with different superconductor systems, the BCS theory has subsequently become inadequate to fully explain how superconductivity is occurring.

Brian Josephson

Another significant theoretical advancement came in 1962 when Brian D. Josephson (above), a graduate student at Cambridge University, predicted that electrical current would flow between 2 superconducting materials - even when they are separated by a non-superconductor or insulator. His prediction was later confirmed and won him a share of the 1973 Nobel Prize in Physics. This tunneling phenomenon is today known as the "Josephson effect" and has been applied to electronic devices such as the SQUID, an instrument capabable of detecting even the weakest magnetic fields. (Below SQUID graphic courtesy Quantum Design.)

The 1980's were a decade of unrivaled discovery in the field of superconductivity. In 1964 Bill Little of Stanford University had suggested the possibility of organic (carbon-based) superconductors. The first of these theoretical superconductors was successfully synthesized in 1980 by Danish researcher Klaus Bechgaard of the University of Copenhagen and 3 French team members. (TMTSF)₂PF₆ had to be cooled to an incredibly cold 1.2K transition temperature (known as Tc) and subjected to high pressure to superconduct. But, its mere existence proved the possibility of "designer" molecules - molecules fashioned to perform in a predictable way.

Then, in 1986, a truly breakthrough discovery was made in the field of superconductivity. Alex Müller and Georg Bednorz (above), researchers at the IBM Research Laboratory in Rüschlikon, Switzerland, created a brittle ceramic compound that superconducted at the highest temperature then known: 30 K. What made this discovery so remarkable was that ceramics are normally insulators. They don't conduct electricity well at all. So, researchers had not considered them as possible high-temperature superconductor candidates. The Lanthanum, Barium, Copper and Oxygen compound that Müller and Bednorz synthesized, behaved in a not-as-yet-understood way. (Original article printed in Zeitschrift für Physik Condensed Matter, April 1986.) The discovery of this first of the superconducting copper-oxides (cuprates) won the 2 men a Nobel Prize the following year. It was later found that tiny amounts of this material were actually superconducting at 58 K, due to a small amount of lead having been added as a calibration standard - making the discovery even more noteworthy.

Müller and Bednorz' discovery triggered a flurry of activity in the field of superconductivity. Researchers around the world began "cooking" up ceramics of every imaginable combination in a quest for higher and higher Tc's. In January of 1987 a research team at the University of Alabama-Huntsville substituted Yttrium for Lanthanum in the Müller and Bednorz molecule and achieved an incredible 92 K Tc. For the first time a material (today referred to as YBCO) had been found that would superconduct at temperatures warmer than liquid nitrogen - a commonly available coolant. Additional milestones have since been achieved using exotic - and often toxic - elements in the base perovskite ceramic. The current class (or "system") of ceramic superconductors with the highest transition temperatures are the mercuric-cuprates. The first synthesis of one of these compounds was achieved in 1993 at the University of Colorado and by the team of A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott of Zurich, Switzerland. The world record Tc of 138 K is now held by a thallium-doped, mercuric-cuprate comprised of the elements Mercury, Thallium, Barium, Calcium, Copper and Oxygen. The Tc of this ceramic superconductor was confirmed by Dr. Ron Goldfarb at the National Institute of Standards and Technology-Colorado in February of 1994. Under extreme pressure its Tc can be coaxed up even higher - approximately 25 to 30 degrees more at 300,000 atmospheres.

The first company to capitalize on high-temperature superconductors was Illinois Superconductor (today known as ISCO International), formed in 1989. This amalgam of government, private-industry and academic interests introduced a depth sensor for medical equipment that was able to operate at liquid nitrogen temperatures (~ 77K).

In recent years, many discoveries regarding the novel nature of superconductivity have been made. In 1997 researchers found that at a temperature very near absolute zero an alloy of gold and indium was both a superconductor and a natural magnet. Conventional wisdom held that a material with such properties could not exist! Since then, over a half-dozen such compounds have been found. Recent years have also seen the discovery of the first high-temperature superconductor that does NOT contain any copper (2000), and the first all-metal perovskite superconductor (2001).

Also in 2001 a material that had been sitting on laboratory shelves for decades was found to be an extraordinary new superconductor. Japanese researchers measured the transition temperature of magnesium diboride at 39 Kelvin - far above the highest Tc of any of the elemental or binary alloy superconductors. While 39 K is still well below the Tc's of the "warm" ceramic superconductors, subsequent refinements in the way MgB₂ is fabricated have paved the way for its use in industrial applications. Laboratory testing has found MgB₂ will outperform NbTi and Nb₃Sn wires in high magnetic field applications like MRI.

Though a theory to explain high-temperature superconductivity still eludes modern science, clues occasionally appear that contribute to our understanding of the exotic nature of this phenomenon. In 2005, for example, Superconductors.ORG discovered that increasing the weight ratios of alternating planes within the layered perovskites can often increase Tc significantly. This has led to the discovery of more than 35 new high-temperature superconductors, including a candidate for a new world record.

The most recent "family" of superconductors to be discovered is the "pnictides". These iron-based superconductors were first observed by a group of Japanese researchers in 2006. Like the high-Tc copper-oxides, the exact mechanism that facilitates superconductivity in them is a mystery. However, with Tc's over 50K, a great deal of excitement has resulted from their discovery.

Researchers do agree on one thing: discovery in the field of superconductivity is as much serendipity as it is science. Stay tuned!

[For additional, more-obscure history, visit the "Atypical" and "Type 2"pages.]

[Last page rev: December 2008]

source : http://superconductors.org/History.htm

How to Build Your Own Solar Cell

Step 1 - Stain the Titanium Dioxide with the Natural Dye: Stain the white side of a glass plate which has been coated with titanium dioxide (TiO). This glass has been previously coated with a transparent conductive layer (SnO), as well as a porous TiO film. Crush fresh (or frozen) blackberries, raspberries, pomegranate seeds, or red Hibiscus tea in a tablespoon of water. Soak the film for 5 minutes in this liquid to stain the film to a deep red-purple color. If both sides of the film are not uniformly stained, then put it back in the juice for 5 more minutes. Wash the film in ethanol and gently blot it dry with a tissue.


Step 2 - Coat the Counter Electrode: The solar cell needs both a positive and a negative plate to function. The positive electrode is called the counter electrode and is created from a "conductive" SnO coated glass plate. A Volt - Ohm meter can be used to check which side of the glass is conductive. When scratched with a finger nail, it is the rough side. The "non-conductive" side is marked with a "+." Use a pencil lead to apply a thin graphite (catalytic carbon) layer to the conductive side of plate's surface.


Steps 3 & 4 - Add the Electrolyte and Assemble the Finished Solar Cell: The Iodide solution serves as the electrolyte in the solar cell to complete the circuit and regenerate the dye. Place the stained plate on the table so that the film side is up and place one or two drops of the iodide/iodine electrolyte solution on the stained portion of the film. Then place the counter electrode on top of the stained film so that the conductive side of the counter electrode is on top of the film. Offset the glass plates so that the edges of each plate are exposed. These will serve as the contact points for the negative and positive electrodes so that you can extract electricity and test your cell.

Use the two clips to hold the two electrodes together at the corner of the plates. The output is approximately 0.43 V and 1 mA/cm2 when the cell is illuminated in full sun through the TiO side. Source : http://www.solideas.com/solrcell/english.html

Modified Plants May Yield More Biofuel

ScienceDaily (Dec. 26, 2008) — Plants, genetically modified to ease the breaking down of their woody material, could be the key to a cheaper and greener way of making ethanol, according to researchers who add that the approach could also help turn agricultural waste into food for livestock.

Lignin, a major component of woody plant material,, is woven in with cellulose and provides plants with the strength to withstand strong gusts of wind and microbial attack. However, this protective barrier or "plastic wall" also makes it harder to gain access to the cellulose.

"There is lots of energy-rich cellulose locked away in wood," said John Carlson, professor of molecular genetics, Penn State. "But separating this energy from the wood to make ethanol is a costly process requiring high amounts of heat and caustic chemicals. Moreover, fungal enzymes that attack lignin are not yet widely available, still in the development stage, and not very efficient in breaking up lignin."

Researchers have previously tried to get around the problem by genetically decreasing the lignin content in plants. However, this can lead to a variety of problems -- limp plants unable to stay upright, and plants more susceptible to pests.

"Trying to engineer trees without lignin is like trying to engineer boneless chicken," said Ming Tien, professor of biochemistry, Penn State. "It just doesn't make sense."

Carlson, Tien and postdoctoral associate Haiying Liang use a different genetic approach. Instead of decreasing the lignin content, they are trying to modify the connections in lignin, without compromising either the biosynthesis of lignin or the structural rigidity of the plant.

The Penn State geneticists and biochemists took a gene from beans and engineered it into a poplar tree. This gene produces a protein that inserts itself between two lignin molecules when the lignin polymer is created.

"Now we have a lignin polymer with a protein stuck in between," explained Carlson, who, along with Tien and Liang, has filed a provisional patent on the approach. "When that occurs, it creates a type of lignin that is not much different in terms of strength than normal lignin, but we can break open the lignin polymer by using enzymes that attack proteins rather than enzymes that attack lignin."

These enzymes that attack proteins are already used widely in the laundry detergent industry and are commercially readily available, added Carlson.

The genetic modification does not appear to weaken the plants, and the transformation may have turned them into more efficient sources of ethanol.

"When we looked at the first generation of modified plants we noticed that the lignin content has not changed," said Tien, whose work is funded by the U.S. Department of Energy. "We haven't done a fitness test yet but we did see an increase in the yield of sugars for converting into ethanol."

The researchers may also have stumbled on an unexpected side benefit.

One of the problems with forage crops such as ryegrass and clover is that they have too much lignin, which can cause ruminants like cows to get sick. Their digestive enzymes go into overdrive to break down the lignin, creating a lot of gas and digestion problems for the animals.

"All animals produce enzymes in their digestive process that break down amino acids and small proteins that can be absorbed by the intestine," said Carlson. "If this technology were to be transferred to alfalfa or hay or such cattle feed, it might make it easier for the cows to break down the lignin through their own enzymes."

Carlson added that the technology could potentially be transferred to other biomass crops and even help turn agricultural waste products found on farms into animal feed. But the modified plants will require federal approval before they can be commercialized.

http://www.sciencedaily.com/releases/2008/12/081222163051.htm