{"id":171,"date":"2010-10-07T23:48:11","date_gmt":"2010-10-08T03:48:11","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=171"},"modified":"2010-10-08T00:11:00","modified_gmt":"2010-10-08T04:11:00","slug":"some-facts","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=171","title":{"rendered":"Some facts"},"content":{"rendered":"<p>1) For any number N=Sigma(p_i), i=1..k, p_i are prime factors of N with any prime number cp, cp is not factor of N, there is:<br \/>\n    Mod(cp^LCM(p_i-1, i=1..k), N)=1.<br \/>\n    for some case, Mod(cp^(LCM(p_i-1, i=1..k)\/2), N)= +\/-1<\/p>\n<p>    Special: when N is prime, Mod(cp^(N-1), N)=1<\/p>\n<p>    For example:<br \/>\nIn[3]:= FactorInteger[66855224152]<br \/>\nOut[3]= {{2, 3}, {19, 1}, {1549, 1}, {283949, 1}}<br \/>\nIn[5]:= LCM[18,1548,283948]<br \/>\nOut[5]= 109887876<br \/>\nIn[6]:= Mod[3^109887876,66855224152]<br \/>\nOut[6]= 1<br \/>\nIn[8]:= Mod[5^109887876,66855224152]<br \/>\nOut[8]= 1<br \/>\nIn[9]:= Mod[5^(109887876\/2),66855224152]<br \/>\nOut[9]= 1<br \/>\nIn[10]:= Mod[7^109887876,66855224152]<br \/>\nOut[10]= 1<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1) For any number N=Sigma(p_i), i=1..k, p_i are prime factors of N with any prime number cp, cp is not factor of N, there is: Mod(cp^LCM(p_i-1, i=1..k), N)=1. for some case, Mod(cp^(LCM(p_i-1, i=1..k)\/2), N)= +\/-1 Special: when N is prime, Mod(cp^(N-1), N)=1 For example: In[3]:= FactorInteger[66855224152] Out[3]= {{2, 3}, {19, 1}, {1549, 1}, {283949, 1}} [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6],"tags":[],"class_list":["post-171","post","type-post","status-publish","format-standard","hentry","category-to-entertain-myself","category-looking-for-a-megaprime","post-blog"],"_links":{"self":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/171","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=171"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/171\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=171"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=171"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=171"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}