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

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

IMPORTANCE OF PRIME NUMBERS

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 ?

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:

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

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

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.

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.

Copyright © 2000 by Monica Yuskaitis

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

– Cross out 1; it is not prime.

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

– Leave 2; cross out multiples of 2

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 Total of digits = 15 3 divides evenly into is a multiple of 3

– Leave 3; cross out multiples of 3

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.

– Leave 5; cross out multiples of 5

– Leave 7; cross out multiples of 7

–Leave 11; cross out multiples of 11

All the numbers left are prime

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

Similarly if we want to find the years of our century i.e 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.

st Century

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

 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.

 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.