{"id":254,"date":"2026-08-29T18:28:35","date_gmt":"2026-08-29T18:28:35","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=254"},"modified":"2026-09-03T18:39:57","modified_gmt":"2026-09-03T18:39:57","slug":"element-5-and-set-115t5t","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/29\/element-5-and-set-115t5t\/","title":{"rendered":"Element 5 and set 1+(1+5^t)*(5^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>5<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2217<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>5<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1+(1+5^t)*(5^{t})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">We will be using generating element 5<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The groups generated by 5 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 5 next<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using prefix NPh 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 5 mod N when N of the form 1+(1+5^t)*(5^t) is 3\u00d7t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When t=1 we have P=1+6\u00d75 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 5 has 3 elements<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>The elements in <strong>NPh3<\/strong> are { 5, 25, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If <strong>NPh3<\/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 5 is 25 mod P<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">For 651 from t=2 the 6 elements in <strong>NPh6<\/strong> are as shown next<br>   { 5, 25, 125, 625, 521, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 15751 from t=3 the 9 elements in <strong>NPh9<\/strong> are as shown next<br>   { 5, 25, 125, 625, 3125, 15625, 15121, 12601, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 391251 from t=4 the 12 elements in <strong>NPh12<\/strong> are as shown next<br>   { 5, 25, 125, 625, 3125, 15625,<br>   78125, 390625, 388121, 375601, 313001, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 9768751 from t=5 the 15 elements in <strong>NPh15<\/strong> are as shown next<br>   { 5, 25, 125, 625, 3125, 15625,<br>   78125, 390625, 1953125, 9765625, 9753121, 9690601,<br>   9378001, 7815001, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>For 244156251 from t=6 the 18 elements in <strong>NPh18<\/strong> are as shown next<br>   { 5, 25, 125, 625, 3125, 15625,<br>   78125, 390625, 1953125, 9765625, 48828125, 244140625,<br>   244078121, 243765601, 242203001, 234390001, 195325001, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 6103593751 from t=7 the 21 elements in <strong>NPh21<\/strong> are as shown next<br>   { 5, 25, 125, 625, 3125, 15625,<br>   78125, 390625, 1953125, 9765625, 48828125, 244140625,<br>   1220703125, 6103515625, 6103203121,<br>   6101640601, 6093828001, 6054765001,<br>   5859450001, 4882875001, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 152588281251 from t=8 the 24 elements in <strong>NPh24<\/strong> are as shown next<br>   { 5, 25, 125, 625, 3125,<br>   15625, 78125, 390625, 1953125, 9765625,<br>   48828125, 244140625, 1220703125, 6103515625, 30517578125,<br>   152587890625, 152586328121, 152578515601,<br>   152539453001, 152344140001, 151367575001, <br>   146484750001, 122070625001, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For t=9 the 27 elements in <strong>NPh27<\/strong> are not tabulated here<br>For t=10 the 30 elements in <strong>NPh30<\/strong> are not tabulated here<br>For t=11 the 33 elements in <strong>NPh33<\/strong> are not tabulated here<br>For t=12 the 36 elements in <strong>NPh36<\/strong> are not tabulated here<br>For t=13 the 39 elements in <strong>NPh39<\/strong> are not tabulated here<br>For t=14 the 42 elements in <strong>NPh42<\/strong> are not tabulated here<br>For t=15 the 45 elements in <strong>NPh45<\/strong> are not tabulated here<br>For t=16 the 48 elements in <strong>NPh48<\/strong> are not tabulated here<br>For t=17 the 51 elements in <strong>NPh51<\/strong> are not tabulated here<br>For t=18 the 54 elements in <strong>NPh54<\/strong> are not tabulated here<br>For t=19 the 57 elements in <strong>NPh57<\/strong> are not tabulated here<br>For t=20 the 60 elements in <strong>NPh60<\/strong> are not tabulated here<\/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=\"758\" height=\"62\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forPlus5times5toT.png\" alt=\"sturdy element notation for element 5 in enclosing set 1+(1+5^t)*(5^t)\" class=\"wp-image-255\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forPlus5times5toT.png 758w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forPlus5times5toT-300x25.png 300w\" sizes=\"auto, (max-width: 758px) 100vw, 758px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the space We will be using generating element 5 The groups generated by 5 have low order [ a lot lower than n-1 ] Tabulating the elements of groups generated by 5 next Using prefix NPh 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-254","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\/254","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=254"}],"version-history":[{"count":2,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/254\/revisions"}],"predecessor-version":[{"id":311,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/254\/revisions\/311"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=254"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=254"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=254"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}