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Prime Numbers in Java Integrated qrcode in Java Prime Numbers

4.3 Prime Numbers using barcode writer for none control to generate, create none image in none applications. ASP.NET and Visual Web Developer A sophisticat none for none ed result concerning the number of primes is the important "Prime Number Theorem," which was proven independently by Jacques Hadamard and Charles-Jean de la Vallee-Pousin in 1896. It gives an approximation to the function Jr(x), which is the number of primes p < x. For example, 7r(5) = 3 since there are three primes p < 5.

We have earlier noted that there are 25 primes p < 100; thus, 7r(100) = 25. Let 7r(x) denote the number of primes p < x. Then.

wr (X) lim x--* ocx/In x equivalently,. mr(x) -. x Inx Students unce rtain about limits might wish a translation! First of all, we say "is asymptotic to" for -. So the second statement of the theorem reads "7r(x) is asymptotic to x/lnx," from which we infer that 7r(x) is approximately equal to x/ In x for large x, the approximation getting better and better as x grows. Setting x = 100, Theorem 4.

3.13 asserts that the number of primes p < 100 is roughly 100/ In 100 ; 21.715.

Note that r(100) 100/ In 100 25 21.715. [The symbol t none for none means "approximately."] Setting x = 1,000,000, the theorem says that the number of primes under I million is roughly 1,000,000/ In 1,000,000 t 72,382. In fact, there are 78,498 such primes.

Note that 7r (1,000,000) "_____ 1,000,000/ In 1,000,000. __ _ _ _ _ _ _ _. 78,498 72,382. As x gets lar ger, the fraction xlInx gets closer and closer to 1. 2T(x) On June 23, 1993, at a meeting at the Isaac Newton Institute in Cambridge, England, Andrew Wiles of Princeton University announced a proof of "Fermat"s Last Theorem," arguably the most famous open mathematics problem of all time. For any integer n > 2, the equation a" + bV = c" has no nonzero integer solutions a, b, c.

Notice that it is sufficient to prove this theorem just for the case that n is a prime. For example, if we knew that a3 + b3 = c3 had no integral solutions, then neither would A3n + B3n = C3 n. If the latter had a solution, so would the former,.

with a = An, b = B", c =C. When we first learned the Theorem of Pythagoras for right-angled triangles, we discovered that there are many triples a, b, c of integers which satisfy a2 +b2 = c2 ; for example, 3, 4, 5 and 5, 12, 13, but are there triples of integers a, b, c satisfying a3 + b3 = C3 or a7 + b 7 = c 7 or an + b = cn for any values ofn. except n = 2 Pierre de Fer mat was notorious for scribbling ideas in the margins of whatever he was reading. In 1637, he wrote in the margin of Diophantus"s book Arithmetic that he had found a "truly wonderful" proof that a" + bn = c" had no solutions in the positive integers for n > 2, but that there was insufficient space to. 122 4 The Integers write it down . Truly wonderful it must have been because for over 350 years, mathematicians were unable to find a proof, though countless many tried! Amateur and professional mathematicians alike devoted years and even lifetimes to working on Fermat"s Last Theorem. The theorem owes its name, by the way, to the fact that it is the last of the many conjectures made by Fermat during his lifetime to have resisted resolution.

By now, most of Fermat"s unproven suggested theorems have been settled (and found to be true). In 1983, Gerd Faltings proved that, for each n > 2, the equation a" + bV = c could have at most a finite number of solutions. While this was a remarkable achievement, it was a long way from showing that this finite number was zero.

Then, in 1985, Kenneth Ribet of Berkeley showed that Fermat"s Last Theorem was a consequence of a conjecture first proposed by Yutaka Taniyama in 1955 and clarified by Goro Shimura in the 1960s. It was a proof of the Shimura-Taniyama conjecture which Andrew Wiles announced on June 23, 1993, a truly historic day in the world of mathematics. The months following this announcement were extremely exciting as mathematicians all over the world attempted to understand Wiles"s proof.

Not unexpectedly in such a complex and lengthy argument, a few flaws were found. By the end of 1994, however, Wiles and one of his former graduate students, Richard Taylor, had resolved the remaining issues to the satisfaction of all. Among the many exciting accounts of the history of Fermat"s Last Theorem and of Wiles"s work, we draw special attention to an article by Barry Cipra, "Fermat"s Theorem-at Last!", which was the leading article in What"s Happening in the Mathematical Sciences, Vol.

3 (1995-1996), published by the American Mathematical Society. Faltings himself wrote "The Proof of Fermat"s Last Theorem by R. Taylor and A.

Wiles" for the Notices of the American Mathematical Society, Vol. 42 (1995), No. 7.

In fact, many excellent accounts have been written. We cite just a few. There is one entitled "The Marvelous Proof" by Fernando Q.

Gouvea, which appeared in the American MathematicalMonthly, Vol. 101 (1994), and others by Ram Murty, Notes of the Canadian Mathematical Society, Vol. 25 (September 1993) and by Keith Devlin, Fernando Gouvea, and Andrew Granville, Focus, Vol.

13, Mathematical Association of America (August 1993)..
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