{"id":1194,"date":"2026-05-05T03:25:10","date_gmt":"2026-05-05T03:25:10","guid":{"rendered":"https:\/\/yesrenee.com\/?p=1194"},"modified":"2026-05-09T19:53:12","modified_gmt":"2026-05-09T19:53:12","slug":"the-sieve-of-eratosthenes","status":"publish","type":"post","link":"https:\/\/yesrenee.com\/?p=1194","title":{"rendered":"The Sieve of Eratosthenes"},"content":{"rendered":"\n
\"\"\/<\/figure>\n\n\n\n

That is a sieve.<\/p>\n\n\n\n

I usually use sieves to sift my cake flour. Sifting cake flour means the bulky stuff is forced through these small holes. Thus, the flour ends up fine; finer and softer than sand.<\/p>\n\n\n\n

Why must I sift my cake flour? So the finished cake is light as a feather and soft as silk and fluffy as a cotton fiber.<\/p>\n\n\n\n

Math also needs sifting sometimes. Why have a bulky, indigestible rock when we can have something as fluffy as a chiffon cake?<\/p>\n\n\n\n

2000 years ago, Eratosthenes thought so too. (Actually, chiffon cake was invented 1800 years later, but let’s not discuss that here.)<\/p>\n\n\n\n

How many such primes are there between two given numbers $a$ and $b$?<\/p>\n\n\n\n

You could solve this with brute force. Test all the numbers between or equal to $a$ and $b$.<\/p>\n\n\n\n

But who wants too? If you had two large numbers, $1000000000$ and $2000000000$, the process would take years!<\/p>\n\n\n\n

Eratosthenes knew that too.<\/p>\n\n\n\n

He made his own sieve.<\/p>\n\n\n\n

Explain what Eratosthenes’ sieve is.<\/h2>\n\n\n\n

If you think about it, any composite integer is equal to a prime multiplied by any other integer (prime or composite, it doesn’t matter, as long as it’s more than 1).<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

$1$ is definitely not prime. $2$ is prime, of course.<\/p>\n\n\n\n

Let’s take all the multiples of $2$ within these boundaries… that are not $2$ itself, of course!<\/p>\n\n\n\n

$4$, $6$, $8$ and so on, until $100$. Let’s cross them off.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The next number that isn’t crossed off is 3, so let’s cross off all of 3’s multiples.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The next number that hasn’t been crossed off is 5, so let’s cross off all of its multiples. And let’s keep going until we can’t cross any more.<\/p>\n\n\n\n

We’re done with:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Have you notcied? This removes all of the composite numbers!<\/p>\n\n\n\n

Now it’s easy to count how many primes there are.<\/p>\n\n\n\n

How can I implement this in coding?<\/h2>\n\n\n\n

First, let’s try to solve this question:<\/p>\n\n\n\n

How many primes are between $a$ and $b$, inclusive?<\/p>\n\n\n\n

First, let’s try a simple case. $a = 1$ and $b = 10$.<\/p>\n\n\n\n

Let’s create a vector array and call it<\/p>\n","protected":false},"excerpt":{"rendered":"

That is a sieve. I usually use sieves to sift my cake flour. Sifting cake flour means the bulky stuff is forced through these small holes. Thus, the flour ends up fine; finer and softer than sand. Why must I sift my cake flour? So the finished cake is light as a feather and soft […]<\/p>\n","protected":false},"author":1,"featured_media":1221,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[46,55,22,54],"class_list":["post-1194","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-daily","tag-coding","tag-history","tag-math","tag-number-theory"],"_links":{"self":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1194"}],"version-history":[{"count":7,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1194\/revisions"}],"predecessor-version":[{"id":1220,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1194\/revisions\/1220"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/media\/1221"}],"wp:attachment":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1194"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1194"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}