Sieve of Eratosthenes

Around 240 BC, the Greek mathematician, geographer, and astronomer, Eratosthenes of Cyrene, discovered a way of finding all the prime numbers up to a given limit.

It's called the sieve of Eratosthenes, and it's an ancient algorithm that finds prime numbers.

How it works is first you choose the limit up to which you want to explore.

Let's say you want to find all the primes up to 100.

The root of 100 is 10, so we only need to cross out multiples of 2, 3, 5, and 7. and you'll find that even just these four numbers will eliminate every non-prime up to 100.

So there you have it.

The sieve of Eratosthenes sieves a whole range of numbers to leave only the primes.

This method is intriguing because you're not so much finding the primes as finding what isn't prime, which is very effective because there are more composite numbers in the given range than there are primes.

The sieve of Eratosthenes is a timeless, elegant method for finding all the prime numbers in a large number set.

It doesn't involve adding new numbers or creating new shapes.

Instead, it uses a clever process of subtraction and filtering.

The Pythagoreans considered the even number of which the duad was the prototype to be indefinite and feminine.

The odd numbers are divided by a mathematical contrivance called the Sieve of Eratosthenes into three general classes: incomposite, composite, and incomposite composite.

The incomposite numbers are those which have no divisor other than themselves and unity, such as three, five, seven, eleven, thirteen, seventeen, nineteen, twenty-three, twenty-nine, thirty-one, thirty-seven, forty-one, forty-three, forty-seven, and so forth.

For example, seven is divisible only by seven, which goes into itself once, and unity, which goes into seven seven times.

Thus, every fourth number is evenly odd.

Each of the even odd numbers may be divided once as two, which becomes two ones and cannot be divided further, or six, which becomes two threes and cannot be divided further.

The Sieve of Eratosthenes.

Redrawn from Taylor's Theoretic Arithmetic.