{"id":9544531,"date":"2007-05-03T11:44:00","date_gmt":"2007-05-03T11:44:00","guid":{"rendered":"http:\/\/su.blog.bunty.tv\/2007\/05\/03\/Math-Games-Evil-Numbers\/"},"modified":"2007-11-25T23:56:50","modified_gmt":"2007-11-25T23:56:50","slug":"Math-Games-Evil-Numbers","status":"publish","type":"post","link":"http:\/\/su.blog.bunty.tv\/?p=9544531","title":{"rendered":"Math Games: Evil Numbers"},"content":{"rendered":"
Stumbleupon Review<\/a> of :
\n\thttp:\/\/www.maa.org\/editorial\/mathgames\/mathgames_10_04_04.html<\/a><\/a>\n<\/div>\n

From the page: “Evil numbers are easy to find, as one might expect with an arbitrarily defined property. Interestingly, π is doubly evil. Many numbers aren’t evil. For example, e isn’t evil. The important sums are 665 and 668 after 141 decimal digits — e goes over 666 without hitting it. Chris Lomont notes that you can take every 15th digit of e, and show that e is somewhat evil. The farthest he had to reach for any constant was with Catalan’s constant, which wasn’t evil until he took every 28th digit.<\/p>\n

It appears that most numbers are at least somewhat evil. With enough fiddling, almost anything is likely to be findable.<\/p>\n

How common are evil numbers? (number of non-zero digits \/ sum of digits) gives a good estimate. For a base 10 number, the estimate gives 9\/(1+2+3+4+5+6+7+8+9) = 20% = 1\/5. Thus, any random number has a 1 in 5 chance of having this property.<\/p>\n

A fifth is an estimate. Chris Lomont found a recursion that could give an exact value:<\/p>\n

p[1] = 1\/9;
\n p[k_ \/; k <10] := p[k] = 1\/9 + 1\/9 Sum[p[j], {j, 1, k – 1}];
\n p[k_ \/; k > 9] := p[k] = 1\/9 Sum[p[k – d], {d, 1, 9}];
\n Timing[Table[p[100 n], {n, 1, 10}]][[1]]<\/p>\n

The exact probability of 666 being hit is the following:<\/p>\n

(large number edited out as it was breaking my blog)<\/p>\n

\nWhich is approximately .2000000000000000000000000000000000000000000000000000000000000002166222683713523944720537405934866672. That’s very close to 1\/5. The third term in the continued fraction expansion of p[666] is 184653222869167741981875869102352405779668736930185085305398884.<\/p>\n

The probability of evilness converges quickly to 1\/5, but it does so in a highly oscillatory way. Note that each term is being multiplied by 1.243n.<\/p>\n

<\/a><\/p>\n

Figure 1. ListPlot[Table[(1\/5 – p[n]) (1243\/1000)^n, {n, 30, 900}]]<\/p>\n

Another method Chris Lomont discovered for calculating evilness involved the series expansion of (1-t9)\/(t10-10t+9). In this series expansion, coefficient n is identical to p[n].<\/p>\n

It seemed like there should be a closed form for all of this, but it was beyond Chris’s ability to calculate. Also, I wasn’t able to calculate the evilness probability for the continued fraction of a random irrational number. This calls for wisdom. If anyone has insight, let them calculate the equations.”\t<\/p>\n","protected":false},"excerpt":{"rendered":"

Stumbleupon Review of : http:\/\/www.maa.org\/editorial\/mathgames\/mathgames_10_04_04.html From the page: “Evil numbers are easy to find, as one might expect with an arbitrarily defined property. Interestingly, π is doubly evil. Many numbers aren’t evil. For example, e isn’t evil. The important sums … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":""},"categories":[343],"tags":[247],"_links":{"self":[{"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=\/wp\/v2\/posts\/9544531"}],"collection":[{"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9544531"}],"version-history":[{"count":0,"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=\/wp\/v2\/posts\/9544531\/revisions"}],"wp:attachment":[{"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9544531"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9544531"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/su.blog.bunty.tv\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9544531"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}