{"id":269,"date":"2026-09-02T19:28:16","date_gmt":"2026-09-02T19:28:16","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=269"},"modified":"2026-09-03T18:40:25","modified_gmt":"2026-09-03T18:40:25","slug":"element-2-and-set-112t2t","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/02\/element-2-and-set-112t2t\/","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><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 NPb 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 2 mod N when N of the form 1+(1+2^t)*(2^t) is 3\u00d7t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When t=3 we have P=1+9\u00d78 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 9 elements<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The elements in <strong>NPb9<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 55, 37, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If <strong>NPb9<\/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 37 mod P<\/li>\n\n\n\n<li>Inverse of 4 is 55 mod P<\/li>\n\n\n\n<li>Inverse of 8 is 64 mod P<\/li>\n\n\n\n<li>Inverse of 16 is 32 mod P<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">From 273 for t=4 the 12 elements in <strong>NPb12<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 256, 239, 205, 137, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>From 1057 for t=5 the 15 elements in <strong>NPb15<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,<br>   991, 925, 793, 529, 1 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From 4161 for t=6 the 18 elements in <strong>NPb18<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,<br>   2048, 4096, 4031, 3901, 3641, 3121, 2081, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From 16513 for t=7 the 21 elements in <strong>NPb21<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128,<br>   256, 512, 1024, 2048, 4096, 8192, 16384,<br>   16255, 15997, 15481, 14449, 12385, 8257, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From 65793 for t=8 the 24 elements in <strong>NPb24<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 256,<br>   512, 1024, 2048, 4096, 8192, 16384, 32768, 65536,<br>   65279, 64765, 63737, 61681, 57569, 49345, 32897, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For t=9 the 27 elements in <strong>NPb27<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128, 256, 512,<br>   1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144,<br>   261631, 260605, 258553, 254449, 264241, 229825, 196993,<br>   131329, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For t=10 the 30 elements in <strong>NPb30<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128,<br>   256, 512, 1024, 2048, 4096, 8192,<br>   16384, 32768, 65536, 131072, 262144,<br>   524288, 1048576, 1047551, 1045501, 1041401, 1033201,<br>   1016801, 984001, 918401, 787201, 524801, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For t=11 the 33 elements in <strong>NPb33<\/strong> are not tabulated here<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For t=12 the 36 elements in <strong>NPb36<\/strong> are as shown next<br>   { 2, 4, 8, 16, 32, 64, 128,<br>   256, 512, 1024, 2048, 4096, 8192,<br>   16384, 32768, 65536, 131072, 262144,<br>   524288, 1048576, 2097152, 4194304, 8388608, 16777216,<br>   16773119, 16764925, 16748537, 16715761, 16650209, 16519105,<br>   16256897, 15732481, 14683649, 12585985, 8390657, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For t=13 the 39 elements in <strong>NPb39<\/strong> are not tabulated here<br>For t=14 the 42 elements in <strong>NPb42<\/strong> are not tabulated here<br>For t=15 the 45 elements in <strong>NPb45<\/strong> are not tabulated here<br>For t=16 the 48 elements in <strong>NPb48<\/strong> are not tabulated here<br>For t=17 the 51 elements in <strong>NPb51<\/strong> are not tabulated here<br>For t=18 the 54 elements in <strong>NPb54<\/strong> are not tabulated here<br>For t=19 the 57 elements in <strong>NPb57<\/strong> are not tabulated here<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For t=20 the 60 elements in <strong>NPb60<\/strong> are as shown next<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">{ 2, 4, 8, 16, 32,<br>64, 128, 256, 512, 1024,<br>2048, 4096, 8192, 16384, 32768,<br>65536, 131072, 262144, 524288, 1048576,<br>2097152, 4194304, 8388608, 16777216, 33554432,<br>67108864, 134217728, 268435456, 536870912, 1073741824,<br>2147483648, 4294967296, 8589934592,<br>17179869184, 34359738368, 68719476736,<br>137438953472, 274877906944, 549755813888,<br>1099511627776, 1099510579199, 1099508482045,<br>1099504287737, 1099495899121, 1099479121889,<br>1099445567425, 1099378458497, 1099244240641, 1098975804929, 1098438933505, 1097365190657, <br>1095217704961, 1090922733569, 1082332790785,<br>1065152905217, 1030793134081, 962073591809,<br>824634507265, 549756338177, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">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=\"860\" height=\"63\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2nonprothBfor3to20.png\" alt=\"Sturdy element notation for 2 for set 1+(1+2^t)*(2^t) for t=3,4,5,6,7,8,9,10,12,20\" class=\"wp-image-271\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2nonprothBfor3to20.png 860w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2nonprothBfor3to20-300x22.png 300w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2nonprothBfor3to20-764x56.png 764w\" sizes=\"auto, (max-width: 860px) 100vw, 860px\" \/><\/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 NPb 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-269","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\/269","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=269"}],"version-history":[{"count":3,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/269\/revisions"}],"predecessor-version":[{"id":314,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/269\/revisions\/314"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=269"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=269"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}