Primes are the building blocks of whole numbers, and important to the world because their odd mathematical properties make them perfect for our current uses.

Introduction:

in this Essay, I would like to write about prime numbers. I am writing on the subject of Mathematics and I have chosen this topic because of personal interests that is I like Mathematics. Besides this I think the topic possesses an enormous importance, too.

the use of primes is important question. Why is that so? What properties do prime numbers have that they make codes very difficult to decipher?

What numbers are called primes? We know that they are only from the set of integers. Why not have prime numbers in other number sets, such as in the set of complex numbers?

Prime number is the number, which is > 1 and can be only divided by itself or one.

History, Facts and Examples

Primes are importance to number theorists because they are the building blocks of whole numbers and very useful in crypography.

A method to determine whether a number is prime or not, is to divide it by prime number that is less than or equal to the square root of that number. If the results of the division is an integer, the original number is not a prime and if not, the number is a prime. A primality test algorithm is an algorithm that is used to test a number whether the number is a prime number or not.

” AKS primality test

The AKS primality test is based upon the equivalence

(x – a)n = (xn – a) (mod n) for a coprime to n, which is true if and only if n is prime. This is a generalization of Fermat’s little theorem extended to polynomials and can easily be proven using the binomial theorem together with the fact that: for all 0 ; k ; n if n is prime. While this equivalence constitutes a primality test in itself, verifying it takes exponential time. Therefore AKS makes use of a related equivalence

(x – a)n = (xn – a) (mod n, x r – 1), which can be checked in polynomial time.

;Fermat's little theorem asserts that if p is prime and 1? a ; p, then

a p -1? 1 (mod p)

In order to test whether p is a prime number or not, one can pick random a's in the interval and check if there is an equality;.

; Solovay-Strassen primality test

For a prime number p and any integer a,

A (p -1)/2 ? (a/p) (mod p)

Where (a/p) is the Legendre symbol. The Jacobi symbol is a generalization of the Legendre symbol to (a/n); where n can be any odd integer. The Jacobi symbol can be computed in time O((log n)²) using Jacobi's generalization of law of quadratic reciprocity.

It can be observed whether or not the congruence

A (n -1)/2 ? (a/n) (mod n) holds for various values of a. This congruence is true for all a's if n is a prime number. (Solovay, Robert M. and Volker Strassen, 1977)

;This test is for a natural number n and in this test, it is also required that the prime factors of n ? 1 should be already known.

If for every prime factor (q) of n ? 1, there exists an integer a less than n and greater than 1 such as

a n -1 ?1 (mod n)

and then

a n -1/q 1 (mod n)

then n is prime. If no such number can be found, n is composite number;. There are several witnesses 'a' for every odd composite n. But, a simple way to generate such an 'a' is known

Euler showed that the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … is divergent.

Facts about prime numbers

73939133 is an amazing prime number. If the last or the digit at the units place is removed, every time you will get a prime number. It is the largest known prime with this property. Because, all the numbers which we get after removing the end digit of the number are also prime numbers. They are as follows: 7393913, 739391, 73939, 7393, 739, 73 and 7. All these numbers are prime numbers. This is a distinct quality of the number 73939133, which any other number does not have.

The only even prime number is 2. 0 and 1 are not considered to be prime numbers. With the exception of 0 and 1, a number is either a prime number or a composite number. A composite number is any number that is bigger than 1 and that is not prime.

The last digit of a prime number greater than 5 can never be 5. Any number greater than 5 whose last digit is 5 can be divided by 5. (Prime Numbers, 2008)

Fermat primality test

Fermat's little theorem asserts that if p is prime and 1? a ; p, then

a p -1? 1 (mod p)

In order to test whether p is a prime number or not, one can pick random a's in the interval and check if there is an equality.

; Solovay-Strassen primality test

For a prime number p and any integer a,

A (p -1)/2 ? (a/p) (mod p)

Jacobi symbol can be computed in time O((log n)²) using Jacobi's generalization of law of quadratic reciprocity.

It can be observed whether or not the congruence

A (n -1)/2 ? (a/n) (mod n) holds for various values of a. This congruence is true for all a's if n is a prime number. (Solovay, Robert M. and Volker Strassen, 1977)

Lucas-Lehmer test

;This test is for a natural number n and in this test, it is also required that the prime factors of n ? 1 should be already known.

If for every prime factor (q) of n ? 1, there exists an integer a less than n and greater than 1 such as

a n -1 ?1 (mod n)

and then

a n -1/q 1 (mod n)

then n is prime. If no such number can be found, n is composite number;.

; Miller-Rabin primality test

If we can find an a such that

ad ? 1 (mod n), and

a2nd -1 (mod n) for all 0 ? r ? s – 1

then 'a' proves the compositeness of n. If not, 'a' is called a strong liar, and n is a strong probable prime to the base a. ;Strong liar; refers to the case where n is composite but yet the equations hold as they would for a prime number.

There are several witnesses 'a' for every odd composite n. But, a simple way to generate such an 'a' is known. Making the test probabilistic is the solution: we choose randomly, and check whether it is a witness for the composite nature of n. If n is composite, majority of the 'a's are witnesses, therefore the test will discover n as a composite number with high probability. (Rabin, 1980)

A probable prime is an integer, which is considered to be probably prime by passing a certain test. Probable primes, which are actually composite (such as Carmichael numbers) are known as pseudoprimes.

Besides these methods, there are other methods also. There is a set of Diophantine equations in 9 variables and one parameter in which the parameter is a prime number only if the resultant system of equations has a solution over the natural numbers. A single formula with the property of all the positive values being prime can be obtained with this method. There is another formula that is based on Wilson's theorem. The number 'two' is generated several times and all other primes are generated exactly once. Also, there are other similar formulas that can generate primes. Some primes are categorized as per the properties of their digits in decimal or other bases. An example is that the numbers whose digits develop a palindromic sequence are palindromic primes, and if by consecutively removing the first digit at the left or the right generates only new prime numbers, a prime number is known as a truncatable prime.

The first 5,000 prime numbers can be known very quickly by just looking at odd numbers and checking each new number (say 5) against every number above it (3); so if 5Mod3 = 0 then it's not a prime number.

History of prime numbers

The most ancient and acknowledged proof for the statement that ;There are infinitely many prime numbers;, is given by Euclid in his Elements (Book IX, Proposition 20). The Sieve of Eratosthenes is a simple, ancient algorithm to identify all prime numbers up to a particular integer. After this, came the modern Sieve of Atkin, which is faster but more complex. The Sieve of Eratosthenes was created in the 3rd century BC by Eratosthenes. Some clues can be found in the surviving records of the ancient Egyptians regarding their knowledge of prime numbers: for example, the Egyptian fraction expansions in the Rhind papyrus have fairly different forms for primes and for composites. But, the first surviving records of the clear study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) include key theorems about primes, counting the fundamental theorem of arithmetic and the infinitude of primes. Euclid also explained how a perfect number is constructed from a Mersenne prime.

Use

Prime number is used in field of encryption. web pages contain personal information and information that you want to keep strictly private such as credit card numbers.

Conclusion

Prime numbers are very unique. They have studied for years, yet only recently have great advances been. Even with great technology we still have a trouble finding them, or any sort of pattern. Prime is very important for computers encryption. I really enjoyed and learned from making this project.