{"id":150,"date":"2026-08-20T20:37:49","date_gmt":"2026-08-20T20:37:49","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=150"},"modified":"2026-09-03T18:38:54","modified_gmt":"2026-09-03T18:38:54","slug":"super-element-2-and-set-1-12t2t1","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/20\/super-element-2-and-set-1-12t2t1\/","title":{"rendered":"Element 2 and set 1+(-1+2^t)*(2^(t+1))"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Consider the Proth 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><mrow><mi>t<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1+(-1+2^t)*(2^{t+1})<\/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 [sub]groups generated by 2 have low order [ a lot lower than n-1 ]<br><\/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 the prefix TNx 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\"><br>From t=2 the elements in <strong>TNx20<\/strong> are as shown next<br> { 2, 4, 8, 16, 7, 14, 3, 6, 12, -1,<br> 23, 21, 17, 9, 18, 11, 22, 19, 13, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br><strong>Conjecture<\/strong>: Order of 2 mod N when N of the form 1+(-1+2^t)*(2^(t+1)) is 4\u00d7(1+2\u00d7t)<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When t=3 we have P=1+7\u00d716 and the set of elements modulo P is a multiplicative group.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The subgroup generated by 2 has 28 elements<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The elements in <strong>TNx28<\/strong> are as shown next<br> { 2, 4, 8, 16, 32, 64, 15, 30, 60, 7, 14, 28, 56, -1,<br> 111, 109, 105, 97, 81, 49, 98, 83, 53, 106, 99, 85, 57, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If <strong>TNx28<\/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 57 mod P<\/li>\n\n\n\n<li>Inverse of 4 is 85 mod P<\/li>\n\n\n\n<li>Inverse of 8 is 99 mod P<\/li>\n\n\n\n<li>Inverse of 16 is 106 mod P<\/li>\n\n\n\n<li>Inverse of 32 is 53 mod P<\/li>\n\n\n\n<li>Inverse of 15 is 98 mod P<\/li>\n\n\n\n<li>Inverse of 30 is 49 mod P<\/li>\n\n\n\n<li>Inverse of 60 is 81 mod P<\/li>\n\n\n\n<li>Inverse of 7 is 97 mod P<\/li>\n\n\n\n<li>Inverse of 14 is 105 mod P<\/li>\n\n\n\n<li>Inverse of 28 is 109 mod P<\/li>\n\n\n\n<li>Inverse of 56 is 111 mod P<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">We could alternatively write the group <strong>TNx28<\/strong> as follows:<br> { 2, 4, 8, 16, 32, 64, 15, 30, 60, 7, 14, 28, 56, -1,<br> -2, -4, -8, -16, -32, -64, -(2^7), -(2^8), -(2^9), -(2^10), -(2^11), <br> -(2^12), -(2^13), 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=4 the 36 elements in <strong>TNx36<\/strong> are as shown next<br> { 2, 4, 8, 16, 32, 64, 128, 256, 31,<br> 62, 124, 248, 15, 30, 60, 120, 240, <strong>-1<\/strong>,<br> 479, 477, 473, 465, 449, 417, 353, 225, 450,<br> 419, 357, 233, 466, 451, 421, 361, 241, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=5 the 44 elements in <strong>TNx44<\/strong> are as shown next<br> { 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,<br> 63, 126, 252, 504, 1008, 31, 62, 124, 248, 496,<br> 992, <strong>-1<\/strong>, 1983, 1981, 1977, 1969, 1953, 1921, 1857, 1729<br> 1473, 961, 1922, 1859, 1733, 1481, 977, 1954, 1923, 1861,<br> 1737, 1489, 993, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=6 the 52 elements in <strong>TNx52<\/strong> are as shown next.<br> { 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,<br> 2048, 4096, 127, 254, 508, 1016, 2032, 4064, 63, 126,<br> 252, 504, 1008, 2016, 4032, <strong>-1<\/strong>, 8063, 8061, 8057, 8049,<br> 8033, 8001, 7937, 7809, 7553, 7041, 6017, 3969, 7938, 7811,<br> 7557, 7049, 6033, 4011, 8002, 7939, 7813, 7561, 7057, 6049,<br> 4033, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=7 the 60 elements in <strong>TNx60<\/strong> are as shown next.<br> { 2, 4, 8, 16, 32, 64,<br> 128, 256, 512, 1024, 2048, 4096,<br> 8192, 16384, 255, 510, 1020, 2040,<br> 4080, 8160, 16320, 127, 254, 508,<br> 1016, 2032, 4064, 8128, 16256, <strong>-1<\/strong>,<br> 32511, 32509, 32505, 32497, 32481, 32449,<br> 32385, 32257, 32001, 31489, 30465, 28417,<br> 24321, 16129, 32258, 32003, 31493, 30473,<br> 28433, 24353, 16193, 32386, 32259, 32005, <br> 31497, 30481. 28449, 24385, 16257, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=8 the 68 elements in <strong>TNx68<\/strong> are as shown next.<br> { 2, 4, 8, 16, 32, 64, <br> 128, 256, 512, 1024, 2048, 4096, <br> 8192, 16384, 32768, 65536, 511, 1022, <br> 2044, 4088, 8176, 16352, 32704, 65408, <br> 255, 510, 1020, 2040, 4080, 8160,<br> 16320, 32640, 65280, <strong>-1<\/strong>, 130559, 130557,<br> 130553, 130545, 130529, 130497, 130433, 130305, <br> 130049, 129537, 128513, 126465, 122369, 114177,<br> 97793, 65025, 130050, 129539, 128517, 126473,<br> 122385, 114209, 97857, 65153, 130306, 130051,<br> 129541, 128521, 126481, 122401, 114241, 97921, 65281, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From t=9 the 76 elements in <strong>TNx76<\/strong> are not tabulated here<br>For t=10 the 84 elements in <strong>TNx84<\/strong> are not tabulated here<br>For t=11 the 92 elements in <strong>TNx92<\/strong> are not tabulated here<br>For t=12 the 100 elements in <strong>TNx100<\/strong> are not tabulated here<br>For t=13 the 108 elements in <strong>TNx108<\/strong> are not tabulated here<br>For t=14 the 116 elements in <strong>TNx116<\/strong> are not tabulated here<br>For t=15 the 124 elements in <strong>TNx124<\/strong> are not tabulated here<br>For t=16 the 132 elements in <strong>TNx132<\/strong> are not tabulated here<br>For t=17 the 140 elements in <strong>TNx140<\/strong> are not tabulated here<br>For t=18 the 148 elements in <strong>TNx148<\/strong> are not tabulated here<br>For t=19 the 156 elements in <strong>TNx156<\/strong> are not tabulated here<br>For t=20 the 164 elements in <strong>TNx164<\/strong> are not tabulated here<br><\/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=\"827\" height=\"70\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/superelement2for2to8minus.png\" alt=\"Super element 2 and set 1+(-1+2^t)*(2^(t+1))\" class=\"wp-image-151\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/superelement2for2to8minus.png 827w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/superelement2for2to8minus-300x25.png 300w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/superelement2for2to8minus-768x65.png 768w\" sizes=\"auto, (max-width: 827px) 100vw, 827px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the Proth space We will be using generating element 2 The [sub]groups generated by 2 have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by 2 next. Using the prefix TNx to avoid clashing with existing letter conventions for groups. Prefer G or S? Replace them in [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6],"tags":[7,8],"class_list":["post-150","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-numbertheory","tag-cyclic","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/150","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=150"}],"version-history":[{"count":7,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/150\/revisions"}],"predecessor-version":[{"id":307,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/150\/revisions\/307"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=150"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=150"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}