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2 5 7 11 3 13 19 23 Prime Time 17 29 31 37 41

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A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and itself.

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The number 1 is a special case which is considered neither prime nor composite. Although the number 1 used to be considered a prime, it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own.

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**With 1 excluded, the smallest prime is therefore 2**

With 1 excluded, the smallest prime is therefore 2. However, since 2 is the only even prime (which, ironically, in some sense makes it the "oddest" prime), it is also somewhat special, and the set of all primes excluding 2 is therefore called the "odd primes."

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The nth prime number is commonly denoted pn, so p1 = 2, p2 = 3, and so on, and may be computed in Mathematica as Prime[n]. The set of primes is sometimes denoted P.

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Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate."

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In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision"

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The fundamental theorem of arithmetic states that any positive integer (greater than 1) can be represented in exactly one way as a product of primes. 12 = 22·3 96= 299 =

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The simplest method of finding factors is so-called "direct search factorization" (a.k.a. trial division). In this method, all possible factors are systematically tested using trial division to see if they actually divide the given number. It is practical only for very small numbers. More general (and complicated) methods include the elliptic curve factorization method and number field sieve factorization method.

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Sieve of Eratosthenes (who also calculated the circumference of the (round) earth). Pick the first prime. Cross off every multiple of that prime. Pick the next prime (the next number not crossed off). Cross off every multiple thereof Continue ad infinitum Wolfram Demonstration

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Infinitude of Primes Proved by Euclid, Book IX, proposition 20: Prime numbers are more than any assigned multitude of prime numbers. Prove by contradiction: Assume P is the largest prime… then multiply: 2x3x5x7x11x…P. Add 1. Any divisor will leave a remainder of 1, therefore there is no largest prime P.

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**Euclid’s Proof Theorem. Proof.**

There are more primes than found in any finite list of primes. Proof. Call the primes in our finite list p1, p2, ..., pr. Let P be any common multiple of these primes plus one (for example, P = p1p2...pr+1). Now P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p. Notice p can not be any of p1, p2, ..., pr, otherwise p would divide 1, which is impossible. So this prime p is some prime that was not in our original list. Either way, the original list was incomplete.

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Note that what is found in this proof is another prime--one not in the given initial set. There is no size restriction on this new prime, it may even be smaller than some of those in the initial set.

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Prime Curiosities With the exception of 2 and 3, all primes are of the form p = 6n ± 1, i.e., p = 1, 5(mod6)

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The function that gives the number of primes less than or equal to a number n is denoted p(n) and is called the prime counting function. The theorem giving an asymptotic form for p(n) is called the prime number theorem.

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p(n) and pn are inverse functions, so p(pn) = n for all positive integers and pp(n) = n iff n is a prime number.

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Mersenne Prime A Mersenne prime is a Mersenne number, i.e., a number of the form Mn = 2n – 1 that is prime. In order for Mn to be prime, n must itself be prime. This is true since for composite n with factors r and s , n=rs. Therefore, 2n - 1 can be written as 2rs - 1 , which is a binomial number that always has a factor (2r – 1). The first few Mersenne primes are 3, 7, 31, 127, 8191, corresponding to indices 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ...

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Twin Primes While Hardy and Wright note that "the evidence, when examined in detail, appears to justify the conjecture," and Shanks states even more strongly, "the evidence is overwhelming," Hardy and Wright also note that the proof or disproof of conjectures of this type "is at present beyond the resources of mathematics." Arenstorf (2004) published a purported proof of the conjecture. Unfortunately, a serious error was found in the proof. As a result, the paper was retracted and the twin prime conjecture remains fully open.

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Twin Prime Conjecture There are an infinite number of twin primes p, p+2 Sophie Germain primes (p & 2p+1) are closely related. The Hardy-Littlewood conjecture states that the number of twin primes x approaches… Where P2 is the twin prime constant, ~

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Bertrand’s Postulate For n > 1, there is always one prime p such that n < p < 2n. Proved independently by Pafnuty Chebychev Srinivasa Ramanujan Paul Erdös

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**The Prime Number Theorem**

The prime number theorem gives an asymptotic form for the prime counting function p(n) , which counts the number of primes less than some integer n .

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Gauss (Gauß) At age 15 had access to a table (by Lambert) of primes up to a million. Also had a table of logs. Looking at the pair, he said… Primzahlen unter a(=∞) a/la Or, “the number of primes under a (as a approaches infinity) is approximately

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**Legendre’s Improvement**

Brought Gauss’s approximation up towards the true number of primes.

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**Logarithmic Integral Gauss later refined his result to**

Calculate Li(100) and Li(10,000)

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**For small n, it had been checked and always found that p(n) < Li(n)**

For small n, it had been checked and always found that p(n) < Li(n). As a result, many prominent mathematicians, including no less than both Gauss and Riemann, conjectured that the inequality was strict. To everyone's surprise, this conjecture was refuted when Littlewood (1914) proved that the inequality reverses infinitely often for sufficiently large n.

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**Chebyshev put limits on the ratio**

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**Hadamard's proof of the prime number theorem depends on the simple trigonometric inequality**

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**Distribution of primes**

Wolfram Demonstration Project: Distribution of Primes Note that Gauss’ approximation overestimates p(n) for small values of n. Note also the “pattern” of primes in the numeric array at the left.

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The function p (x), the prime-counting function, gives the number of primes less than or equal to x. The plots shown are of the differences between p (x) and the approximations to p (x) due to Gauss, Legendre, and Riemann.

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Zeta (z) Function “The zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity…The main interest comes from trying to improve the Prime Number Theorem, i.e. getting better estimates for the distribution of the prime numbers.”

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**Zeta (z) Function (Summation form)**

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**Product Form of Zeta Function**

kp is the pth prime.

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Riemann Hypothesis Prove the Riemann Hypothesis and win a million bucks! Peter Sarnak’s description

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Riemann Hypothesis Essentially, the Riemann Hypothesis says that every zero of z(s) lies on the line x = ½. Song Another variation says that z(1/2 + it)=0

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Möbius Function m(n) The Zeta function is connected to the Möbius function by the following equation:

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**Quantum Physics & Riemann Zeros**

In a chance meeting over tea at Princeton, physicist Freeman Dyson and number theorist Hugh Montgomery happened to talk about what they were each working on. Dyson was looking at the energy levels of electrons in heavy atoms, Montgomery was looking at the zeros of the zeta function. The patterns were the same.

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**Moments of z on the critical line**

At the RHI conference in Seattle in 1995, some mathematicians and physicists said that there is a factor, gk in a formula for moments of z that match eigenvalues of atoms. They knew the first two factors, and after some work found the third. The fourth (and formula) were found by 1998.

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**Moments of z on the critical line**

g1=1 g2=2 g3=42 g4=24,024

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MATH 224 – Discrete Mathematics

MATH 224 – Discrete Mathematics

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