Prime Numbers

Opracowały:Małgorzata MaciorMartyna Owoc

What is a prime number ?

DEFINITION:A natural number p ≥2 is called prime if and only if the only natural numbers which divide p are 1 and p. A natural number n>1 which is not prime is called composite. Thus, n>1 is composite if n=ab where a and b are natural numbers with 1<a,b<n.

Some first prime numbers : 2, 3, 5, 7, 11, 13, 17, 19, …

Euklides, about 365-300 B.C.

Euklides proved, that the set of prime numbers is infinite. He used a proof by contradiction.

THE SIEVE OF ERATOSTHENESTo find all the prime numbers less than or equal to a

given natural number n,

• Continue until all numbers less than or equal to √n have been circled or crossed out.

• list all the integers from 2 to n,

• Circle 2 and then cross out all multiples of 2 in the list,

• Circle 3, the first number not yet crossed out or circled , and then cross out all multiples of 3,

• Circle 5, the first number not yet crossed out or circled , and then cross out all multiples of 5,

• At the general stage, circle the first number which is neither

crossed out nor circled and cross out its multiples,

Properties of Prime Numbers

THE N2 + 1 CONJECTUREThere are infinitely many primes of the form N2 +1.

INFINITELY MANY PRIMES THEOREMThere are infinitely many prime numbers.

THE TWIN PRIMES CONJECTUREThere are infinitely many prime numbers p such that p+2 is also prime.

GOLDBACH’S CONJECTUREEvery even number n≥4 is a sum of two primes.

Given any natural number n>1, there exists prime p such that p|n.

If a natural number n>1 is not prime, then n is divisible by some prime number p≤√n .

TYPES OF PRIMES

Mersenne Primes Twin Primes Quadruple Primes Isolated Primes Mirror Primes Sophie Germain Primes Fermat Primes

TYPES OF PRIMESMersenne Primes If an-1 is prime for some numbers a≥2 and n≥2,

then a must equal 2 and n must be a prime. Primes of the form 2p-1 are called Mersenne primes.

The first few Mersenne primes are:22-1 =3, 23-1=7, 25-1=31.

Twin Primes Two prime numbers the difference of which is two, are called twin primes.Examples of pairs of twin primes:

(3, 5) (5, 7) (59, 61)

5&7

Prime quadruplet A prime quadruplet is a set of four primes of the form {p, p+2, p+6, p+8} .For instance:5, 7, 11, 13821, 823, 827, 829

Isolated Primes A prime number is called isolated when the nearest prime number differentiates by at leat 4.

89 157

173

Mirror Primes Two prime numbers are called mirror primes if one is obtained from the other by reversing its digits.Examples:13 and 3117 and 71

13

17 71

31

Sophie Germain PrimesA prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 29 is a Sophie Germain prime and 2 × 29 + 1 = 59 is its associated safe prime.

Fermat primes Another interesting class of prime numbers is the set of so-called Fermat primes, this being prime numbers of the form 22n + 1. For n=0,1,2,3,4, indeed 22n + 1 is prime.F0 = 21 + 1 = 3F1 = 22 + 1 = 5 F2 = 24 + 1 = 17 F3 = 28 + 1 = 257

Counting Primes

The Prime Number Theorem When x is large, the number of primes less than x is approximately equal to x/ln(x). In other words,

1)()ln(

x

x

xx

Where π(x)=#{primes p with p≤x}

Curosity 1In 1914, American mathematician , Derrick Norman Lehmer published for the first time a list of all 664,579 primes less than 10 million . He created this list using the sieve of Eratosthenes .

Curosity 211111111111111111111111- number consisting of 23 ones is a prime number .

Curosity 3The number31415926535897932384626433832795028841formed from the first 38 digits of the decimal expansion of the number ∏ is prime.

Bibliography Edgar G. Goodaire, Michael M. Parmenter, Discrete

Mathematics with graph Theory, Memorial University of Newfoundland Prentice Hall, Upper Saddle River

Joseph H. Silverman, A Friendly Introduction to Number Theory, Brown University, Prentice Hall, Upper Saddle River

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