{"id":266,"date":"2026-09-02T19:06:15","date_gmt":"2026-09-02T19:06:15","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=266"},"modified":"2026-09-03T18:40:08","modified_gmt":"2026-09-03T18:40:08","slug":"element-2-and-set-1-12t2t","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/02\/element-2-and-set-1-12t2t\/","title":{"rendered":"Element 2 and set 1+(-1+2^t)*(2^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><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2217<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1+(-1+2^t)*(2^{t})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">We will be using generating element 2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The groups generated by 2 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 2 next<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using prefix NNa 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\">From t=3 the elements in <strong>NNa18<\/strong> are <br>   { 2, 4, 8, 16, 32, 7, 14, 28, -1,<br>   55, 53, 49, 41, 25, 50, 43, 29, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conjecture<\/strong>: Order of 2 mod N when N of the form 1+(-1+2^t)*(2^t) is 6\u00d7t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When t=4 we have P=1+15\u00d716 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 2 has 24 elements<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The elements in <strong>NNa24<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 15, 30, 60, 120, -1<br>   239, 237, 233, 225, 209, 177, 113, 226, 211, 181, 121, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If <strong>NNa24<\/strong> 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 2 is 121 mod P<\/li>\n\n\n\n<li>Inverse of 4 is 181 mod P<\/li>\n\n\n\n<li>Inverse of 8 is 211 mod P<\/li>\n\n\n\n<li>Inverse of 16 is 226 mod P<\/li>\n\n\n\n<li>Inverse of 32 is 113 mod P<\/li>\n\n\n\n<li>Inverse of 64 is 177 mod P<\/li>\n\n\n\n<li>Inverse of 128 is 209 mod P<\/li>\n\n\n\n<li>Inverse of 15 is 225 mod P<\/li>\n\n\n\n<li>Inverse of 30 is 233 mod P<\/li>\n\n\n\n<li>Inverse of 60 is 237 mod P<\/li>\n\n\n\n<li>Inverse of 120 is 239 mod P<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">We could alternatively write the group <strong>NNa24<\/strong> as follows:<br>  { 2, 4, 8, 16, 32, 64, (2^7), (2^8), (2^9), (2^10), (2^11), -1,<br>  -2, -4, -8, -16, -32, -64, -(2^7), -(2^8), -(2^9), -(2^10), -(2^11), 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=5 the 30 elements in <strong>NNa30<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 256, 512, 31,<br>   62, 124, 248, 496, <strong>-1<\/strong>, 991, 989, 985, 977, 961,<br>   929, 865, 737, 481, 962, 931, 869, 745, 497, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=6 the 36 elements in <strong>NNa36<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,<br>   2048, 63, 126, 252, 504, 1008, 2016, <strong>-1<\/strong>, 4031, 4029,<br>   4025, 4017, 4001, 3969, 3905, 3777, 3521, 3009, 1985, 3970,<br>   2907, 3781, 3529, 3025, 2017, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=7 the 42 elements in <strong>NNa42<\/strong> are as shown next.<br>   { 2, 4, 8, 16, 32, 64, 128,<br>   256, 512, 1024, 2048, 4096, 8192, 127,<br>   254, 508, 1016, 2032, 4064, 8128, <strong>-1<\/strong>,<br>   16255, 16253, 16249, 16241, 16225, 16193, 16129,<br>   16001, 15745, 15233, 14209, 12161, 8065, 16130,<br>   16003, 15749, 15241, 14225, 12193, 8129, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=8 the 48 elements in <strong>NNa48<\/strong> are as shown next.<br>{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,<br>2048, 4096, 8192, 16384, 32768, 255, 510, 1020, 2040, 4080,<br>8160, 16320, 32640, <strong>-1<\/strong>, 65279, 65277, 65273, 65265, 65249, 65217,<br>65153, 65025, 64769, 64257, 63233, 57089, 48897, 32513, 65026,<br>64771, 64261, 63241, 61201, 57121, 48961, 32641, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=9 the 54 elements in <strong>NNa54<\/strong> are as shown next.<br>{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,<br>2048, 4096, 8192, 16384, 32768, 65536, 131072, 511, 1022, 2044,<br>4088, 8176, 16352, 32704, 65408, 130816, <strong>-1<\/strong>, 261631,<br>261629, 261625, 261617, 261601, 261569, 261505, 261377, 261121,<br>260609, 259585, 257537, 253441, 245249, 228865,<br>196097, 130561, 261122, 260611, 259589, 257545,<br>253457, 245281, 228929, 196225, 130817, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=10 the 60 elements in <strong>NNa60<\/strong> are as shown next.<br>{ 2, 4, 8, 16, 32, 64, 128, 256,<br>512, 1024, 2048, 4096, 8192, 16384, 32768, 65536,<br>131072, 262144, 524288, 1023, 2046, 4092, 8184, 16368,<br>32736, 65472, 130944, 261888, 523776, <strong>-1<\/strong>, 1047551, 1047549,<br>1047545, 1047537, 1047521, 1047489, 1047425, 1047297, 1047041, 1046529, 1045505, 1043457, 1039361, 1031169, 1014785,<br>982017, 916481, 785409, 523265, 1046530, 1045507,<br>1043461, 1039369, 1031185, 1014817, 982081,<br>916609, 785665, 523777, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For t=11 the 66 elements in <strong>NNa66<\/strong> are not tabulated here<br>For t=12 the 72 elements in <strong>NNa72<\/strong> are not tabulated here<br>For t=13 the 78 elements in <strong>NNa78<\/strong> are not tabulated here<br>For t=14 the 84 elements in <strong>NNa84<\/strong> are not tabulated here<br>For t=15 the 90 elements in <strong>NNa90<\/strong> are not tabulated here<br>For t=16 the 96 elements in <strong>NNa96<\/strong> are not tabulated here<br>For t=17 the 102 elements in <strong>NNa102<\/strong> are not tabulated here<br>For t=18 the 108 elements in <strong>NNa108<\/strong> are not tabulated here<br>For t=19 the 114 elements in <strong>NNa114<\/strong> are not tabulated here<br>For t=20 the 120 elements in <strong>NNa120<\/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=\"800\" height=\"63\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1minus2nonprothAfor2to10.png\" alt=\"Sturdy element notation for 2 for set 1+(-1+2^t)*(2^t) for t=3,..,10\" class=\"wp-image-267\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1minus2nonprothAfor2to10.png 800w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1minus2nonprothAfor2to10-300x24.png 300w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1minus2nonprothAfor2to10-762x60.png 762w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the space We will be using generating element 2 The groups generated by 2 have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by 2 next Using prefix NNa 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-266","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\/266","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=266"}],"version-history":[{"count":3,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/266\/revisions"}],"predecessor-version":[{"id":312,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/266\/revisions\/312"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=266"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=266"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=266"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}