{"id":157,"date":"2026-08-21T18:33:04","date_gmt":"2026-08-21T18:33:04","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=157"},"modified":"2026-09-03T18:39:10","modified_gmt":"2026-09-03T18:39:10","slug":"super-element-2-and-set-112t2t1","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/21\/super-element-2-and-set-112t2t1\/","title":{"rendered":"Element 2 and set 1+(1+2^t)*(2^(t+1))"},"content":{"rendered":"\n

Consider the Proth space<\/p>\n\n\n\n

1<\/mn>+<\/mo>(<\/mo>1<\/mn>+<\/mo>2<\/mn>t<\/mi><\/msup>)<\/mo>\u2217<\/mo>(<\/mo>2<\/mn>t<\/mi>+<\/mo>1<\/mn><\/mrow><\/msup>)<\/mo><\/mrow>1+(1+2^t)*(2^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

We will be using generating element 2<\/p>\n\n\n\n

The [sub]groups generated by 2 have low order [ a lot lower than n-1 ]
<\/p>\n\n\n\n

Tabulating the elements of groups generated by 2 next.<\/p>\n\n\n\n

Using the prefix TPv to avoid clashing with existing letter conventions for groups.
Prefer G or S? Replace them in your local copy.<\/p>\n\n\n\n


When t=2 we have P=1+5\u00d78 and the set of elements modulo P is a multiplicative group.<\/p>\n\n\n\n


The subgroup generated by 2 has 20 elements
<\/p>\n\n\n\n

The elements in TPv20<\/strong> are as shown next
{ 2, 4, 8, 16, 32, 23, 5, 10, 20, -1<\/strong>
39, 37, 33, 25, 9, 18, 36, 31, 21, 1 }<\/p>\n\n\n\n

If TPv20<\/strong> is really a group then we need inverses so let us document those next.<\/p>\n\n\n\n

    \n
  • Inverse of 2 is 21 mod P<\/li>\n\n\n\n
  • Inverse of 4 is 31 mod P<\/li>\n\n\n\n
  • Inverse of 8 is 36 mod P<\/li>\n\n\n\n
  • Inverse of 16 is 18 mod P<\/li>\n\n\n\n
  • Inverse of 32 is 9 mod P<\/li>\n\n\n\n
  • Inverse of 23 is 25 mod P<\/li>\n\n\n\n
  • Inverse of 5 is 33 mod P<\/li>\n\n\n\n
  • Inverse of 10 is 37 mod P<\/li>\n\n\n\n
  • Inverse of 20 is 39 mod P<\/li>\n<\/ul>\n\n\n\n

    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)<\/p>\n\n\n\n

    From t=3 the 28 elements in TPv28<\/strong> are as shown next
    { 2, 4, 8, 16, 32, 64, 128, 111, 77, 9,
    18, 36, 72, -1<\/strong>, 143, 141, 137, 129, 113, 81,
    17, 34, 68, 136, 127, 109, 73, 1 }<\/p>\n\n\n\n

    We could alternatively write the group TPv28<\/strong> as follows:
    { 2, 4, 8, 16, 32, 64, 15, 30, 60, 7, 14, 28, 56, -1,
    -2, -4, -8, -16, -32, -64, -(2^7), -(2^8), -(2^9), -(2^10), -(2^11),
    -(2^12), -(2^13), 1 }<\/p>\n\n\n\n

    From t=4 the 36 elements in TPv36<\/strong> are as shown next
    { 2, 4, 8, 16, 32, 64, 128, 256, 512, 479,
    413, 281, 17, 34, 68, 136, 272, -1<\/strong>, 543, 541,
    537, 529, 513, 481, 417, 289, 33, 66, 132, 264,
    528, 511, 477, 409, 273, 1 }<\/p>\n\n\n\n

    From t=5 the 44 elements in TPv44<\/strong> are as shown next
    { 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
    2048, 1983, 1853, 1593, 1073, 33, 66, 132, 264, 528,
    1056, -1<\/strong>, 2111, 2109, 2105, 2097, 2081, 2049, 1985, 1857,
    1601, 1089, 65, 130, 260, 520, 1040, 2080, 2047, 1981,
    1849, 1585, 1057, 1 }<\/p>\n\n\n\n

    From t=6 the 52 elements in TPv52<\/strong> are as shown next.
    { 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
    2048, 4096, 8192, 8063, 7805, 7289, 6257, 4193, 65, 130,
    260, 520, 1040, 2080, 4160, -1<\/strong>, 8319, 8317, 8313, 8305,
    8289, 8257, 8193, 8065, 7809, 7297, 6273, 4225, 129, 258,
    516, 1032, 2064, 4128, 8256, 8191, 8061, 7801, 7281, 6241,
    4161, 1 }<\/p>\n\n\n\n

    From t=7 the 60 elements in TPv60<\/strong> are as shown next.
    { 2, 4, 8, 16, 32, 64,
    128, 256, 512, 1024, 2048, 4096,
    8192, 16384, 32768, 32511, 31997, 30969,
    28913, 24801, 16577, 129, 258, 516,
    1032, 2064, 4128, 8256, 16512, -1<\/strong>,
    33023, 33021, 33017, 33009, 32993, 32961,
    32897, 32769, 32513, 32001, 30977, 28929,
    24833, 16641, 257, 514, 1028, 2056,
    4112, 8224, 16448, 32896, 32767, 32509,
    31993, 30961, 28897, 24769, 16513, 1 }<\/p>\n\n\n\n

    From t=8 the 68 elements in TPv68<\/strong> are as shown next.
    { 2, 4, 8, 16, 32, 64,
    128, 256, 512, 1024, 2048, 4096,
    8192, 16384, 32768, 65536, 131072, 130559,
    129533, 127481, 123377, 115169, 98753, 65921,
    257, 514, 1028, 2056, 4112, 8224,
    16448, 32896, 65792, -1<\/strong>, 131583, 131581,
    131577, 131569, 131553, 131521, 131457, 131329,
    131073, 130561, 129537, 127489, 123393, 115201,
    98817, 66049, 513, 1026, 2052, 4104,
    8208, 16416, 32832, 65664, 131328, 131071,
    130557, 129529, 127473, 123361, 115137, 98689, 65793, 1 }<\/p>\n\n\n\n

    From t=9 the 76 elements in TPv76<\/strong> are not tabulated here
    For t=10 the 84 elements in TPv84<\/strong> are not tabulated here
    For t=11 the 92 elements in TPv92<\/strong> are not tabulated here
    For t=12 the 100 elements in TPv100<\/strong> are not tabulated here
    For t=13 the 108 elements in TPv108<\/strong> are not tabulated here
    For t=14 the 116 elements in TPv116<\/strong> are not tabulated here
    For t=15 the 124 elements in TPv124<\/strong> are not tabulated here
    For t=16 the 132 elements in TPv132<\/strong> are not tabulated here
    For t=17 the 140 elements in TPv140<\/strong> are not tabulated here
    For t=18 the 148 elements in TPv148<\/strong> are not tabulated here
    For t=19 the 156 elements in TPv156<\/strong> are not tabulated here
    For t=20 the 164 elements in TPv164<\/strong> are not tabulated here<\/p>\n\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    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 TPv to avoid clashing with existing letter conventions for groups.Prefer G or S? Replace them in your […]<\/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-157","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\/157","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=157"}],"version-history":[{"count":7,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/157\/revisions"}],"predecessor-version":[{"id":308,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/157\/revisions\/308"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}