{"id":291,"date":"2026-09-03T17:46:04","date_gmt":"2026-09-03T17:46:04","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=291"},"modified":"2026-09-30T17:00:24","modified_gmt":"2026-09-30T17:00:24","slug":"conjectures-1","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/03\/conjectures-1\/","title":{"rendered":"Conjectures 1"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Assorted conjectures 1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conjecture<\/strong>: Factors of N when t is prime and N of the form 1+(-1+2^t)*(2^(t+1)) and 1+2*t is prime must be either 5 or of the form 1+4*k*(1+2*t) where k is a positive integer<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Example for t=11 pr=8384513 we have the factors 277 and 30269 where 1+2*t is 23<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">277=1+4*3 *23<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">30269=1+4*7*47*23<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">8384513==277*30269<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conjecture<\/strong>: Order of b mod N when N of the form<\/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><mi>b<\/mi><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><msup><mi>b<\/mi><mi>t<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">1+(\u22121+b^t)\n\u00d7b^t <\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">is 6\u00d7t where b and t are positive integers &gt;=2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conjecture<\/strong>: Order of b mod N when N of the form<\/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><mi>b<\/mi><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><msup><mi>b<\/mi><mi>t<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">1+(1+b^t)\n\u00d7b^t <\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">is 3\u00d7t where b and t are positive integers &gt;=2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conjecture<\/strong>: There are an infinite number of cyclic groups having finite order.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Posts on this site provide an abundant source of cyclic groups.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"80\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2minus3probableZ-1024x80.png\" alt=\"\" class=\"wp-image-292\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2minus3probableZ-1024x80.png 1024w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2minus3probableZ-300x24.png 300w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2minus3probableZ-764x60.png 764w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2minus3probableZ.png 1223w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/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><mn>16224<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mn>16227<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>P<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">1+(-1+2^{16224})\u00d7(2^{16227}) ~ ~ is ~ ~ Prime<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"758\" height=\"95\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap28conjectureZstronger4denominator.jpeg\" alt=\"\" class=\"wp-image-425\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap28conjectureZstronger4denominator.jpeg 758w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap28conjectureZstronger4denominator-300x38.jpeg 300w\" sizes=\"auto, (max-width: 758px) 100vw, 758px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"88\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-2-1024x88.png\" alt=\"\" class=\"wp-image-295\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-2-1024x88.png 1024w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-2-300x26.png 300w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-2-767x66.png 767w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-2.png 1221w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/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><mn>639<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mn>642<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>P<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">1+(1+2^{639})\u00d7(2^{642}) ~ ~ is ~ ~ Prime<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"743\" height=\"90\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap29conjectureCstronger8denominator.jpeg\" alt=\"\" class=\"wp-image-426\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap29conjectureCstronger8denominator.jpeg 743w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap29conjectureCstronger8denominator-300x36.jpeg 300w\" sizes=\"auto, (max-width: 743px) 100vw, 743px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Assorted conjectures 1 Conjecture: Factors of N when t is prime and N of the form 1+(-1+2^t)*(2^(t+1)) and 1+2*t is prime must be either 5 or of the form 1+4*k*(1+2*t) where k is a positive integer Example for t=11 pr=8384513 we have the factors 277 and 30269 where 1+2*t is 23 277=1+4*3 *23 30269=1+4*7*47*23 8384513==277*30269 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,5,6],"tags":[7,8],"class_list":["post-291","post","type-post","status-publish","format-standard","hentry","category-abstractalgebra","category-mathematics","category-numbertheory","tag-cyclic","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/291","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=291"}],"version-history":[{"count":3,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/291\/revisions"}],"predecessor-version":[{"id":428,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/291\/revisions\/428"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}