{"id":111,"date":"2026-08-14T19:23:30","date_gmt":"2026-08-14T19:23:30","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=111"},"modified":"2026-09-02T20:23:32","modified_gmt":"2026-09-02T20:23:32","slug":"defining-a-super-element","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/14\/defining-a-super-element\/","title":{"rendered":"Defining a sturdy element"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The set of elements modulo 35 is a set<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Working modulo 35 does not give you a group because we can find zero divisors and those are not invertible.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Multiples of 5 and 7 are zero divisors because they are divisors of 35.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There is no inverse for the elements 5, 7, 10, 14, 15, 20, 21, 25, 28,30 in that set<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Listing the elements not on that list gives us a subset consisting of 24 elements as follows<br>   { 1,2,3,4,6,8,9,11,12,13,16,17,18,19,22,23,24,26,27,29,31,32,33,34 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Does the zero divisor 5 generate a group?<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Powering of 5 just gives us a series of non-invertible results<br>   { 5, 25, 20, 30, 10, 15 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Our generated results do not include the identity element 1 or similar and as we noted there are no inverses.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Does the element 6 generate a group?<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We obtain a cyclic group having 2 elements { 6, 1 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Does the zero divisor 7 generate a group?<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Successive powering of 7 just gives us a series of non-invertible results { 7, 14, 28, 21 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Our generated results do not include the identity element 1 or similar and as we noted there are no inverses.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Does the element 3 generate a multiplicative group?<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We obtain a cyclic group having 12 elements<br>   { 3, 9, 27, 11, 33, 29, 17, 16, 13, 4, 12, 1 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Does the element 11 generate a multiplicative group?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We obtain a cyclic group having 3 elements { 11, 16, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Next we attempt to define what a super element is<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>(i) A <strong>sturdy element<\/strong> requires context<br>[ a set in which it\u2019s properties are special ]<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(ii) A <strong>sturdy element<\/strong> always generates a group [or stronger] from that enclosing context.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(iii) A <strong>sturdy element<\/strong> generates a multiplicative group whose order is<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mo rspace=\"0em\">&lt;<\/mo><mo lspace=\"0em\">=<\/mo><mfrac><mrow><mo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">&lt;= \\frac{(n\u22121) }{2}\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><br>(iv a) A <strong>sturdy element<\/strong> shares the context set with an element that generates a maximal group (n-1) elements (depending on the value of our source variable [t])<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>(iv b) A <strong>sturdy element<\/strong> shares the context set with an element that generates a near maximal group (n-1-zero divisors based adjustment) elements (depending on the value of our source variable [t])<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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\">the element -1 + 2^t is a sturdy element<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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\">the element 392+t is not a sturdy element<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Setting t=5 we obtain 397 for 392+t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Working modulo 1985 we obtain the following from successive powering of 397<br>   { 397, 794, 1588, 1191, 397, \u2026 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That subset cannot be a group because it contains a zero divisor in 397<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That series of elements is not a group so our element 392+t fails property (ii) and is therefore not a sturdy element.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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><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\">the element 1 + 2^t is a sturdy element.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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><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\">the element 105+t is not a sturdy element.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Setting t=4 we obtain 109 for 105+t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Working modulo 545 we obtain the following from successive powering of 109   { 109, 436, 109 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>That subset cannot be a group as it contains a zero divisor in 109<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That series of elements is not a group so our element 105+t fails property (ii) and is therefore not a sturdy element.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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>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><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+3^t)*(3^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">the element -1 + 3^t is a sturdy element<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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>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><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+3^t)*(3^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">the element 38+t is not a sturdy element<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Setting t=5 we obtain 43 for 38+t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Working modulo 2107 we obtain the following from successive powering of 43<br>   { 43, 1849, 1548, 1247, 946, 645, 344, 43, \u2026 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That subset cannot be a group as it contains a zero divisor in 43<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That series of elements is not a group so our element 38+t fails property (ii) and is therefore not a sturdy element.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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><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+3^t)*(3^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">the element 1 + 3^t is a sturdy element<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the set<\/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><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+3^t)*(3^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">the element 25406+t is not a sturdy element<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Setting t=5 we obtain 25411 for 25406+t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Working modulo 177877 we obtain the following from successive powering of 25411   { 25411 }<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That subset cannot be a group as it contains a zero divisor in 25411<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That series of elements is not a group so our element 24506+t fails property (ii) and is therefore not a sturdy element.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Author: Gary Wright 2026<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This post appears as the first chapter in a <a href=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsuperChap0to30.pdf\">draft book<\/a> and is available as <a href=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/chap1definesturdy.pdf\">a pdf<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>1+2^t is a super element for set 1+(1+2^t)*(2^(t+1))<\/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-111","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\/111","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=111"}],"version-history":[{"count":16,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/111\/revisions"}],"predecessor-version":[{"id":282,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/111\/revisions\/282"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=111"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=111"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=111"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}