If, like me, you consider yourself a “serious” ( ™ ) arithmetician, you may also have started spurning at an early age the type of numerical coincidences, sometimes found in dictionaries of remarkable numbers, that state that, for instance, $latex n=5$ is the only integral solution to the equation
\n$latex n^{n-2}=n!+n.$<\/p>\n
However, maybe I shouldn’t be so dismissive. Today, after a discussion with P-O Dehaye, which was certainly serious enough since the Ramanujan $latex \\tau$ function<\/a> featured prominently, the question came to understand the solutions in $latex \\mathbf{Z}\/5\\mathbf{Z}$ of the system This sounds rather innocuous, but something I find rather amusing happens: you can check for yourself that the solutions are either (i) diagonal (all $latex v_i$ are equal) or (ii) a permutation<\/i> of $latex \\mathbf{Z}\/5\\mathbf{Z}$ (i.e., no two $latex v_i$ coincide). Why is that? Well, first both (i) and (ii) do describe solutions. Clearly (( ™ ) again) there are $latex 5^{5-2}=125$ solutions in total; there are $latex 5$ diagonal ones, and $latex 5!=120$ permutations; since Is there a more direct proof?<\/p>\n","protected":false},"excerpt":{"rendered":" If, like me, you consider yourself a “serious” ( ™ ) arithmetician, you may also have started spurning at an early age the type of numerical coincidences, sometimes found in dictionaries of remarkable numbers, that state that, for instance, $latex n=5$ is the only integral solution to the equation $latex n^{n-2}=n!+n.$ However, maybe I shouldn’t … Continue reading Numerical oddities…<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-2381","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/2381","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=2381"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/2381\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=2381"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=2381"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=2381"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n$latex v_0+v_1+v_2+v_3+v_4=0,$
\n$latex v_0^2+v_1^2+v_2^2+v_3^2+v_4^2=0,$
\n(everything modulo $latex 5$).<\/p>\n
\n$latex 5^3=5!+5,$
\nthere can be no other solution. <\/p>\n