{"id":1306,"date":"2026-07-26T21:08:00","date_gmt":"2026-07-26T21:08:00","guid":{"rendered":"https:\/\/yesrenee.com\/?p=1306"},"modified":"2026-07-26T21:08:02","modified_gmt":"2026-07-26T21:08:02","slug":"quadratic-residues","status":"publish","type":"post","link":"https:\/\/yesrenee.com\/?p=1306","title":{"rendered":"Quadratic residues"},"content":{"rendered":"\n

We must have a wonderful gradient background here because square number remainder patterns are beautiful. When you square integers and reduce them modulo n<\/em>, you don’t just get a random number. Insead, square numbers follow predictable patterns<\/strong>. These are called quadratic residues.<\/em> Note the examples presented below.<\/p>\n\n\n\n

\n
\n
Modulus<\/th>Possible square remainders<\/th><\/tr><\/thead>
2<\/td> 0, 1<\/td><\/tr>
3<\/td> 0, 1<\/td><\/tr>
4<\/td> 0, 1<\/td><\/tr>
5<\/td> 0, 1, 4<\/td><\/tr>
6<\/td> 0, 1, 3, 4<\/td><\/tr>
7<\/td> 0, 1, 2, 4<\/td><\/tr>
8<\/td> 0, 1, 4<\/td><\/tr>
9<\/td> 0, 1, 4, 7<\/td><\/tr>
10<\/td> 0, 1, 4, 5, 6, 9<\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Does it make sense? To help with your understanding, let’s pick a number and break it down. Modulo by 10 should be obvious- just find the ones digit- so let’s work with the digit 9. <\/p>\n\n\n\n

\n
\n
Number<\/th>Square<\/th>Remainder modulo 9<\/th><\/tr><\/thead>
0<\/td> 0<\/td> 0<\/td><\/tr>
1<\/td> 1<\/td> 1<\/td><\/tr>
2<\/td> 4<\/td> 4<\/td><\/tr>
3<\/td> 9<\/td> 0<\/td><\/tr>
4<\/td> 16<\/td> 7<\/td><\/tr>
5<\/td> 25<\/td> 7<\/td><\/tr>
6<\/td> 36<\/td> 0<\/td><\/tr>
7<\/td> 49<\/td> 4<\/td><\/tr>
8<\/td> 64<\/td> 1<\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div>\n<\/div>\n\n\n\n

We see that the possible remainders, arranged from least to greatest, are 0, 2, 4 and 7. Now, fill out the same table with a different modulus. I would suggest 3, 6, 7 or 8 because they’re more challenging than numbers like 5 and 10.<\/p>\n\n\n\n

In summary, these patterns are very useful for math, especially contest math and number theory related questions. These patterns lead to greater concepts such as the Legendre Symbol<\/em>, and quadratic reciprocity<\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":"

We must have a wonderful gradient background here because square number remainder patterns are beautiful. When you square integers and reduce them modulo n, you don’t just get a random number. Insead, square numbers follow predictable patterns. These are called quadratic residues. Note the examples presented below. Modulus Possible square remainders 2 0, 1 3 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[86,22,54,85,45],"class_list":["post-1306","post","type-post","status-publish","format-standard","hentry","category-daily","tag-aahwhileiwaswritingthisihadamosquitobitesadface","tag-math","tag-number-theory","tag-patterns","tag-pre-contest-review"],"_links":{"self":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1306","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=1306"}],"version-history":[{"count":2,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1306\/revisions"}],"predecessor-version":[{"id":1308,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1306\/revisions\/1308"}],"wp:attachment":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1306"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1306"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1306"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}