{"id":5450,"date":"2025-12-20T14:50:00","date_gmt":"2025-12-20T18:50:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5450"},"modified":"2026-01-09T03:57:30","modified_gmt":"2026-01-09T07:57:30","slug":"number-theory-challenge-12-20-2025","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5450","title":{"rendered":"Number Theory Challenge &#8211; 12\/20\/2025"},"content":{"rendered":"\n<p>Prove that for all positive integer $n$, $19^{2^n}=1+m\\cdot 2^{n+2}$, where $m$ is a positive odd integer.<a href=\"javascript:toggle_visibility('number-theory-chall-12-20-2025')\">\ud83d\udd11<\/a><\/p>\n<div id=\"number-theory-chall-12-20-2025\" style=\"display:none\"><\/p>\n\n\n\n<p><strong>Proof:<\/strong> We prove it by induction.<\/p>\n\n\n\n<p>Base Case: When $n=1$, we have $$19^{2^n}=361=1+45\\cdot 2^3=1+45\\cdot 2^{n+2}$$ Therefore we prove the case when $n=1$, is true with $m=45$ as an odd integer.<\/p>\n\n\n\n<p>Inductive Step: Assume for $n$, $19^{2^n}=1+m\\cdot 2^{n+2}$, where $m$ is a positive odd integer. Therefore for $n+1$, we have $$19^{2^{n+1}}=(19^{2^n})^2=(1+m\\cdot 2^{n+2})^2=1+2m\\cdot 2^{n+2}+(m\\cdot 2^{n+2})^2$$ $$\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ =1+m\\cdot 2^{n+3}+m^2\\cdot 2^{2n+4}=1+(m+m^2\\cdot 2^{n+1})\\cdot 2^{(n+1)+2}$$<\/p>\n\n\n\n<p>Because $m$ is odd, $m^2\\cdot 2^{n+1}$ is even, therefore $m+m^2\\cdot 2^{n+1}$ is odd. Therefore, we prove the case for $n+1$.<\/p>\n\n\n\n<p>By induction, the problem is proved.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Prove that for all positive integer $n$, $19^{2^n}=1+m\\cdot 2^{n+2}$, where $m$ is a positive odd integer.\ud83d\udd11 Proof: We prove it by induction. Base Case: When $n=1$, we have $$19^{2^n}=361=1+45\\cdot 2^3=1+45\\cdot 2^{n+2}$$ Therefore we prove the case when $n=1$, is true &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=5450\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[12],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5450"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5450"}],"version-history":[{"count":16,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5450\/revisions"}],"predecessor-version":[{"id":5503,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5450\/revisions\/5503"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5450"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5450"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5450"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}