Observations on a theorem of Fermat and others on looking at prime numbers

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It is so easy to calculate a p -1 that most elementary primality tests are built using a version of Fermat's Little Theorem rather than Wilson's Theorem. As usual Fermat did not provide a proof this time saying "I would send you the demonstration, if I did not fear its being too long" [ Burton80 , p79].

Euler first published a proof in , but Leibniz left virtually the same proof in an unpublished manuscript from sometime before Start by listing the first p -1 positive multiples of a : a , 2 a , 3 a , Multiply all these congruences together and we find a.

Divide both side by p -1! Then, to punctuate the result, he added:. First proposed by the 17th-century French jurist and spare-time mathematician Pierre de Fermat, it had remained unproven for more than years.

Proof of Fermat's Little Theorem

Wiles, a professor at Princeton University, had worked on the problem, alone and in secret in the attic of his home, for seven years. Now he was unveiling his proof. His announcement electrified his audience—and the world. The story appeared the next day on the front page of The New York Times. Gap, the clothing retailer, asked him to model a new line of jeans, though he demurred.


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Out of the ruins of these failures rose deep theories that opened up vast new areas of mathematics. To say that Thomas Robert Malthus was unpopular would be putting it mildly. His 19th-century contemporary Percy Shelley, the revered poet, called him a eunuch and a tyrant. Once a proof is proposed, it must be checked and verified before it is accepted as valid.

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When Wiles submitted his page proof to the prestigious journal Inventiones Mathematicae , its editor divvied up the manuscript among six reviewers. One of them was Nick Katz, a fellow Princeton mathematician. Katz would email Wiles, who would provide a fix.

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And when Wiles took a closer look, he saw that Katz had found a crack in the mathematical scaffolding. At first, a repair seemed straightforward. But as Wiles picked at the crack, pieces of the structure began falling away. To his mounting horror, Wiles realized that his mistake was more than a minor miscalculation.

At best, it was a gap—left by a faulty anchor point he would have to replace with materials not yet invented. But what if it were a chasm, an irreparable—and humiliating—error that would send the whole thing crashing into the void? Fermat proposed it in while studying Arithmetica , a collection of algebraic problems compiled by the Ancient Greek mathematician Diophantus of Alexandria. The book includes a discussion of the Pythagorean Theorem, which describes a relation among the sides of a right triangle.

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As you may recall, it states that the square of the hypotenuse the side opposite the right angle equals the sum of the squares of the other two sides. The simplest example is the set 3, 4, and 5. Another is 5, 12, and So is , , and Fermat asked a natural next question: Can you find similar triples in higher dimensions?

The answer, Fermat said, is no.

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And, it seemed, he never elaborated elsewhere. After his death, his son Samuel published a new edition of Arithmetica that included all of the notes that Fermat had made in its margins.

Observations On A Theorem Of Fermat And Others On Looking At Prime Numbers

These observations, typically stated without proof, posed an irresistible challenge. Within a few years, readers had confirmed every one—except the statement about higher-dimension Pythagorean triples. Science academies offered handsome prizes.


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  8. And the notoriety of the theorem grew as even the best mathematical minds made limited progress in solving it. No one could prove he was wrong either. Wiles had told few people that the proof mathematicians had been awaiting for centuries was now in danger of crumbling at his feet. Their curiosity, however, brought other rewards.


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    The real number 6, for example, always equals 2 x 3. Aside from reordering the factors 3 x 2 , no other product will work. To restore unique factorization to complex numbers, Kummer invented algebraic objects he called ideals, which proved important in the subsequent development of modern abstract algebra.