NANDAN GOEL 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 ID: 630520
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Slide1
PRIME NUMBERS
PRESENTED BY :
NANDAN GOELSlide2
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 2
P
− 1, WITH
P
A PRIME. THEY ARE CALLED MERSENNE PRIMES IN HIS HONOR.Slide3
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.Slide4
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.Slide5
SIEVE OF ERATOSTHENESSlide6
MODERN PRIMALITY TESTS
AKS primality test
2002
deterministic
O(log
6+
ε
(
n))Baillie PSW primality test1980probabilisticO(log3 n
)
no known counterexamples
Elliptic curve primality proving
1977
deterministic
O(log
5+
ε
(
n
))
heuristically
Fermat primality
t
est
probabilistic
O(
k
· log
2+
ε
(
n
))
fails for Carmichael numbers
Miller
Rabin primality
test
1980
probabilistic
O(
k
· log
2+
ε
(
n
))
error probability 4
−
k
Solovay
S
trassen primalit
y
test
1977
probabilistic
O(
k
· log
3
n
)
error probability 2
−
kSlide7
FERMAT PRIMALITY TEST
FERMAT
‘S THEOREM STATES THAT
N
P
≡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. Slide8
PRIMES IN NATURE
CICADAS: GENUS MAGICICADASlide9
PRIMES IN ART
OLIVIER MESSIAEN
COMMUNICATION WITH ALIENSSlide10
PRIMES:APPLICATION
PUBLIC KEY CRYPTOGRAPHYHASH TABLESLARGE NUMBER GENERATORSSlide11
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