1. PRIME NUMBERS
PRESENTED BY :
NANDAN GOEL

2. HISTORY
THE STUDY OF SURVIVING RECORDS OF EGYPTIANS SHOW THAT THEY HAD KNOWLEDGE OF PRIMES.
THE GREEK MATHEMATICIAN “EUCLID PERFORMED” SOME EXCEPTIONAL WORK .
HIS WORK “EUCLID ELEMENT” CONTAIN IMPORTANT THEOREMS SUCH AS INFINITUDE OF PRIMES AND BASIC THEOREM OF ARITHMETIC.
THE FRENCH MONK “MARIN MERSENNE” LOOKED AT PRIMES OF THE FORM 2P − 1, WITH P A PRIME. THEY ARE CALLED MERSENNE PRIMES IN HIS HONOR.

3. PRIME NUMBER THEOREM
LET Π(X) BE THE PRIME – COUNTING FUNCTION THAT GIVES THE NUMBER OF PRIMES LESS THAN OR EQUAL TO X, FOR ANY REAL NUMBER X. FOR EXAMPLE, Π(10) = 4 BECAUSE THERE ARE FOUR PRIME NUMBERS (2, 3, 5 AND 7) LESS THAN OR EQUAL TO 10. THE PRIME NUMBER THEOREM THEN STATES THAT
X / LOG X IS A GOOD APPROXIMATION TO Π(X), IN THE SENSE THAT THE LIMIT OF THE QUOTIENT OF THE TWO FUNCTIONS Π(X) AND X / LOG X AS X INCREASES WITHOUT BOUND IS 1.

4. TESTING PRIMALITY
THE MOST BASIC METHOD OF CHECKING THE PRIMALITY OF A GIVEN INTEGER N IS CALLED TRIAL DIVISION. THIS ROUTINE CONSISTS OF DIVIDING N BY EACH INTEGER M THAT IS GREATER THAN 1 AND LESS THAN OR EQUAL TO THE SQUARE ROOT OF N. IF THE RESULT OF ANY OF THESE DIVISIONS IS AN INTEGER, THEN N IS NOT A PRIME, OTHERWISE IT IS A PRIME. INDEED, IF N = ab IS COMPOSITE (WITH a AND b ≠ 1) THEN ONE OF THE FACTORS a OR b IS NECESSARILY AT MOST √N.

5. SIEVE OF ERATOSTHENES

6. MODERN PRIMALITY TESTS
AKS primality test
2002
deterministic
O(log6+ε(n))
Baillie PSW primality test
1980
probabilistic
O(log3 n)
no known counterexamples
Elliptic curve primality proving
1977
O(log5+ε(n)) heuristically
Fermat primality test
O(k · log2+ε (n))
fails for Carmichael numbers
Miller Rabin primality test
error probability 4−k
Solovay Strassen primality test
O(k · log3 n)
error probability 2−k

7. FERMAT PRIMALITY TEST
FERMAT‘S THEOREM STATES THAT NP≡N (MOD P) FOR ANY N IF P IS A PRIME.
IF WE HAVE A NUMBER B THAT WE WANT TO TEST FOR PRIMALITY.
THEN WE WORK OUT NB (MOD B) FOR A RANDOM VALUE OF N AS OUR TEST.
A FLAW WITH THIS TEST IS THAT THERE ARE SOME COMPOSITE NUMBERS THAT SATISFY THE FERMAT IDENTITY EVEN THOUGH THEY ARE NOT PRIME.
SINCE THESE COMPOSITE NUMBERS ARE VERY RARE AS COMPARED TO PRIMES SO THIS TEST CAN BE USEFUL FOR PRACTICAL PURPOSES.

8. PRIMES IN NATURE
CICADAS: GENUS MAGICICADA

9. PRIMES IN ART
COMMUNICATION WITH ALIENS
OLIVIER MESSIAEN

10. PRIMES:APPLICATION PUBLIC KEY CRYPTOGRAPHY HASH TABLES
LARGE NUMBER GENERATORS

11. THANK YOU.
ANY QUERIES?