{"id":286,"date":"2026-09-03T11:28:56","date_gmt":"2026-09-03T11:28:56","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=286"},"modified":"2026-09-03T11:28:56","modified_gmt":"2026-09-03T11:28:56","slug":"element-3-and-set-113tx3t","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/03\/element-3-and-set-113tx3t\/","title":{"rendered":"Element 3 and set 1+(1+3^t)\u00d73^t"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Consider the space<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>3<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2217<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>3<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1+(1+3^t)*(3^{t})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">We will be using generating element 3<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The groups generated by 3 have low order [ a lot lower than n-1 ]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Tabulating the elements of groups generated by 3 next<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using prefix NPf to avoid clashing with existing letter conventions for groups.<br>Prefer G or S? Replace them in your local copy.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conjecture<\/strong>: Order of 3 mod N when N of the form 1+(1+3^t)*(3^t) is 3\u00d7t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>When t=3 we have P=1+28\u00d727 and the set of elements modulo P is a multiplicative group.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>The subgroup generated by 3 has 9 elements<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>The elements in NPf9 are as shown next<br>   { 3, 9, 27, 81, 243, 729, 673, 505, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>If NPf9 is really a group then we need inverses so let us document those next.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inverse of 3 is 505 mod P<\/li>\n\n\n\n<li>Inverse of 9 is 673 mod P<\/li>\n\n\n\n<li>Inverse of 27 is 729 mod P<\/li>\n\n\n\n<li>Inverse of 81 is 243 mod P<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">For 6643 from t=4 the 12 elements in <strong>NPf12<\/strong> are as shown next<br>   { 3, 9, 27, 81, 243, 729,<br>   2187, 6561, 6397, 5905, 4429, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 59293 from t=5 the 15 elements in <strong>NPf15<\/strong> are as shown next<br>   { 3, 9, 27, 81, 243,<br>   729, 2187, 6561, 19683, 59049,<br>   58561, 57097, 52705, 39529, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 532171 from t=6 the 18 elements in <strong>NPf18<\/strong> are as shown next<br>   { 3, 9, 27, 81, 243, 729,<br>   2187, 6561, 19683, 59049, 177147, 531441,<br>   529981, 525601, 512461, 473041, 354781, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 4785157 from t=7 the 21 elements in <strong>NPf21<\/strong> are as shown next<br>{ 3, 9, 27, 81, 243, 729, 2187,<br>6561, 19683, 59049, 177147, 531441, 1594323, 4782969,<br>4778593, 4765465, 4726081, 4607929, 4253473, 3190105, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 43053283 from t=8 the 24 elements in <strong>NPf24<\/strong> are as shown next<br>{ 3, 9, 27, 81, 243, 729,<br>2187, 6561, 19683, 59049, 177147, 531441,<br>1594323, 4782969, 14348907, 43046721, 43033597, 42994225,<br>42876109, 42521761, 41458717, 38269585, 28702189, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 387440173 from t=9 the 27 elements in <strong>NPf27<\/strong> are as shown next<br>{ 3, 9, 27, 81, 243, 729,<br>2187, 6561, 19683, 59049, 177147, 531441,<br>1594323, 4782969, 14348907, 43046721, 129140163,<br>387420489, 387381121, 387263017, 386908705, 385845769,<br>382656961, 373090537, 344391265, 258293449, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 3486843451 from t=10 the 30 elements in <strong>NPf30<\/strong> are as shown next<br>{ 3, 9, 27, 81, 243,<br>729, 2187, 6561, 19683, 59049,<br>177147, 531441, 1594323, 4782969, 14348907,<br>43046721, 129140163, 387420489, 1162261467, 3486784401,<br>3486666301, 3486312001, 3485249101, 3482060401, 3472494301,<br>3443796001, 3357701101, 3099416401, 2324562301, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 31381236757 from t=11 the 33 elements in <strong>NPf33<\/strong> are as shown next<br>{ 3, 9, 27, 81, 243,<br>729, 2187, 6561, 19683, 59049,<br>177147, 531441, 1594323, 4782969, 14348907,<br>43046721, 129140163, 387420489, 1162261467, 3486784401,<br>10460353203, 31381059609, 31380705313, 31379642425, 31376453761, 31366887769, 31338189793, 31252095865,<br>30993814081, 30218968729, 27894432673, 20920824505, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 282430067923 from t=12 the 36 elements in <strong>NPf36<\/strong> are as shown next<br>{ 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049,<br>177147, 531441, 1594323, 4782969, 14348907,<br>43046721, 129140163, 387420489, 1162261467, 3486784401,<br>10460353203, 31381059609, 94143178827, 282429536481,<br>282428473597, 282425284945, 282415718989, 282387021121,<br>282300927517, 282042646705, 281267804269, 278943276961,<br>271969695037, 251048949265, 188286711949, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For t=13 the 39 elements in <strong>NPf39<\/strong> are not tabulated here<br>For t=14 the 42 elements in <strong>NPf42<\/strong> are not tabulated here<br>For t=15 the 45 elements in <strong>NPf45<\/strong> are not tabulated here<br>For t=16 the 48 elements in <strong>NPf48<\/strong> are not tabulated here<br>For t=17 the 51 elements in <strong>NPf51<\/strong> are not tabulated here<br>For t=18 the 54 elements in <strong>NPf54<\/strong> are not tabulated here<br>For t=19 the 57 elements in <strong>NPf57<\/strong> are not tabulated here<br>For t=20 the 60 elements in <strong>NPf60<\/strong> are not tabulated here<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Using sturdy element notation we might say the results tabulated in this post are<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"859\" height=\"62\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus3nonprothFfor3to12.png\" alt=\"Sturdy element notation for 3 for set 1+(1+3^t)*(3^t) for t=3,..,12\" class=\"wp-image-289\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus3nonprothFfor3to12.png 859w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus3nonprothFfor3to12-300x22.png 300w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus3nonprothFfor3to12-762x55.png 762w\" sizes=\"auto, (max-width: 859px) 100vw, 859px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the space We will be using generating element 3 The groups generated by 3 have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by 3 next Using prefix NPf to avoid clashing with existing letter conventions for groups.Prefer G or S? Replace them in your local copy. [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,5],"tags":[7,8],"class_list":["post-286","post","type-post","status-publish","format-standard","hentry","category-abstractalgebra","category-mathematics","tag-cyclic","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/286","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/comments?post=286"}],"version-history":[{"count":1,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/286\/revisions"}],"predecessor-version":[{"id":290,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/286\/revisions\/290"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=286"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=286"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}