1. Why are Prime Numbers called prime & Sieve of Eratosthenes Group Members – Umang Chandra Sneh Lata Gupta Shivam Rastogi Rohan Chaudhary Vivek Chaudhary

2. What is a Prime Number? Natural Number > 1 no positive divisors other than 1 and itself Prime Number

4. IMPORTANCE OF PRIME NUMBERS

5. Why are they called Prime – All other numbers (positive integers) are measured by primes, this makes primes first. – We use the English word prime because the ancient Greeks saw them as multiplicatively first, so Billingsley translated Euclid's 'prôtos' as 'prime'. – Other terms used for prime numbers – linear/ simple/ incomposite Prôtos arithmos estin ho monadi monêi metroumenos - Euclid, (The Elements (book 7, definition 11) Meaning- -is measured by a unit alone -are not multiples of other numbers ?

6. Prime numbers are thus the first numbers, the numbers from which the other numbers all arise Thus they are primary numbers and hence are called as:

7. Method to find prime numbers in a given set of natural numbers: Eratosthenes’ Sieve or

8. Eratosthenes (ehr-uh-TAHS-thuh-neez) Eratosthenes was the librarian at Alexandria, Egypt in 200 B.C. Note every book was a scroll.

10. Eratosthenes (ehr-uh-TAHS-thuh-neez) Eratosthenes was a Greek mathematician, astronomer, and geographer. He invented a method for finding prime numbers that is still used today. This method is called Eratosthenes’ Sieve.

11. Eratosthenes’ Sieve A sieve has holes in it and is used to filter out the juice. Eratosthenes’s sieve filters out natural numbers to find the prime numbers.

12. Copyright © 2000 by Monica Yuskaitis

13. Now lets see the main steps how to find the prime numbers using the sieve of Eratosthenes’

14. 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100 1 – Cross out 1; it is not prime.

15. Hint For Next Step Remember all numbers divisible by 2 are even numbers. Like 2,4,6,8,10,12,14………..

16. 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100 2 – Leave 2; cross out multiples of 2

17. Hint For Next Step To find multiples of 3, add the digits of a number; see if you can divide this number evenly by 3; then the number is a multiple of 3. 2 6 7 Total of digits = 15 3 divides evenly into 15 267 is a multiple of 3

18. 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100 3– Leave 3; cross out multiples of 3

19. To find the multiples of 5 look for numbers that end with the digit 0 and 5. Hint For the Next Step 385 is a multiple of 5 & 890 is a multiple of 5 because the last digit ends with 0 or 5.

20. 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100 4– Leave 5; cross out multiples of 5

21. 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100 5– Leave 7; cross out multiples of 7

22. 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100 6–Leave 11; cross out multiples of 11

23. 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100 All the numbers left are prime

24. The Prime Numbers from 1 to 100 are as follows: 2,3,5,7,11,13,17,19, 23,31,37,41,43,47, 53,59,61,67,71,73, 79,83,89,97

25. Similarly if we want to find the years of our century i.e.2000-2099 which are prime numbers we follow the same step by first making the grid of numbers and then crossing the years which are not prime using the above stated method.

26. 2000200120022003200420052006200720082009 2010201120122013201420152016201720182019 2020202120222023202420252026202720282029 2030203120322033203420352036203720382039 2040204120422043204420452046204720482049 2050205120522053205420552056205720582059 2060206120622063206420652066206720682069 2070207120722073207420752076207720782079 2080208120822083208420852086208720882089 2090209120922093209420952096209720982099 21 st Century 2000-2099

27. 2000200120022003200420052006200720082009 2010201120122013201420152016201720182019 2020202120222023202420252026202720282029 2030203120322033203420352036203720382039 2040204120422043204420452046204720482049 2050205120522053205420552056205720582059 2060206120622063206420652066206720682069 2070207120722073207420752076207720782079 2080208120822083208420852086208720882089 2090209120922093209420952096209720982099

28. Thus the prime numbers in this century are: 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099 TOTAL 14 SUCH YEARS WHICH ARE PRIME

29.  It’s one of the educational advantage is that it helps to develop our ability to see and extend pattern.  It is a good method to quickly make a short list of prime no.s.  It is the best intuitive method of finding a list of prime no.s.

30.  It’s disadvantage is that in this method we have to allocate the array at the start and that uses a bunch of memory.  It is a time consuming method because if we want to make a long list of prime no.s then it can take a lot of time.