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Properties of Prime Numbers have been studied as far back as the ancient Greek Civilizations. Be it the mathematicians of <I>Pythagoras'</I> School (500 BC to 300 BC), <I>Euclid</I> in about 300 BC or <I>Eratosthenes</I> in around 200 BC; the Greeks came up with most of the basic and important identities and their proves: <UL> <LI>Pythagoras' Mathematicians came up with the identity of Perfect and Amicable numbers.<BR> <LI>Euclid, in the Book IX of the Elements, proved that there are infinitely many prime numbers.<BR> <LI>Eratosthenes devised a very fast algorithm for calculating primes called the Sieve of Eratosthenes.<BR> </UL> Then there was a very long gap in the history of prime numbers during the Dark Ages. <BR><BR> But the work was again taken up by world famous mathematicians of the likes of <I>Mersenne</I> (Mersenne Primes), <I>Euler</I> (extended Fermat's Little Theorem and found 60 pairs of amicable numbers), <I>Cataldi</I> (proved M<SUP>19</SUP> in 1588) , <I>Fermat</I> (Fermat's Little Theorem) , <I>Legendre</I> and <I>Gauss</I>. <BR><BR> Legendre and Gauss both did extensive calculations on the number of primes in a certain limit, i.e. ƒ(n) is the number of primes in the interval [1,n]. <BR><BR> Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a <I>'chiliad'</I> (a range of 1000 numbers). By the end of his life it is estimated that he had counted all the primes up to about 3 million. <BR><BR> Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about <I>1/log(n)</I>. A Graphical comparison between the actual value of π(n) and estimate ( n / log(n) ) is shown below: <p><center><img SRC="note_55.gif"></center><p> Legendre gave an estimate for π(n) the number of primes n of: <BR><BR> π(n) = n / ( log(n)  1.08366 ) <BR><BR> while Gauss's estimate is in terms of the logarithmic integral: <BR><BR> π(n) = ∫( 1 / log(t) ) dt where the range of integration is from 2 to n. <BR><BR> The statement that the density of primes is 1/log(n) is known as the <I><B><font color="#CC0033">Prime Number Theorem.</B></FONT></I>
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Prof. Ashay Dharwadker