{"id":251,"date":"2026-08-29T17:30:17","date_gmt":"2026-08-29T17:30:17","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=251"},"modified":"2026-08-29T17:32:15","modified_gmt":"2026-08-29T17:32:15","slug":"element-5-and-set-1-15t5t","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/29\/element-5-and-set-1-15t5t\/","title":{"rendered":"Element 5 and set 1+(-1+5^t)*(5^t)"},"content":{"rendered":"\n

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

1<\/mn>+<\/mo>(<\/mo>\u2212<\/mo>1<\/mn>+<\/mo>5<\/mn>t<\/mi><\/msup>)<\/mo>\u2217<\/mo>(<\/mo>5<\/mn>t<\/mi><\/msup>)<\/mo><\/mrow>1+(-1+5^t)*(5^{t})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

We will be using generating element 5
<\/p>\n\n\n\n

The groups generated by 5 have low order [ a lot lower than n-1 ]<\/p>\n\n\n\n

Tabulating the elements of groups generated by 5 next<\/p>\n\n\n\n

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

Conjecture<\/strong>: Order of 5 mod N when N of the form 1+(-1+5^t)*(5^t) is 6\u00d7t<\/p>\n\n\n\n

When t=2 we have P=1+24\u00d725 and the set of elements modulo P is a multiplicative group.<\/p>\n\n\n\n


The multiplicative group generated by 5 has 12 elements<\/p>\n\n\n\n


The elements in NNd12<\/strong> are
{ 5, 25, 125, 24, 120, -1<\/strong>, 596, 576, 476, 577, 481, 1 }
<\/p>\n\n\n\n

If NNd12<\/strong> is really a group then we need inverses so let us document those next.<\/p>\n\n\n\n

    \n
  • Inverse of 5 is 481 mod P<\/li>\n\n\n\n
  • Inverse of 25 is 577 mod P<\/li>\n\n\n\n
  • Inverse of 125 is 476 mod P<\/li>\n\n\n\n
  • Inverse of 24 is 576 mod P<\/li>\n\n\n\n
  • Inverse of 120 is 596 mod P<\/li>\n<\/ul>\n\n\n\n

    We could alternatively write the group NNd12<\/strong> as follows:
    { 5, 25, 125, (5^4), (5^5), -1,
    -5, -25, -125, -(5^4), -(5^5), 1, }<\/p>\n\n\n\n

    For 15501 from t=3 the 18 elements in NNd18<\/strong> are as shown next
    { 5, 25, 125, 625, 3125, 124, 620, 3100, -1<\/strong>,
    15496, 15476, 15376, 14876, 12376, 15377, 14881, 12401, 1 }<\/p>\n\n\n\n

    For 390001 from t=4 the 24 elements in NNd24<\/strong> are as shown next
    { 5, 25, 125, 625, 3125, 15625, 78125, 624,
    3120, 15600, 78000, -1<\/strong>, 389996, 389976, 389876, 389376,
    386876, 374376, 311876, 389377, 386881, 374401, 312001, 1 }<\/p>\n\n\n\n

    For 9762501 from t=5 the 30 elements in NNd30<\/strong> are as shown next
    { 5, 25, 125, 625, 3125, 15625,
    78125, 390625, 1953125, 3124, 15620, 78100,
    390500, 1952500, -1<\/strong>, 9762496, 9762476, 9762376,
    9761876, 9759376, 9746876, 9684376, 9371876, 7809376,
    9759377, 9746881, 9684401, 9372001, 7810001, 1 }<\/p>\n\n\n\n

    For 244125001 from t=6 the 36 elements in NNd36<\/strong> are as shown next
    { 5, 25, 125, 625, 3125, 15625,
    78125, 390625, 1953125, 9765625, 48828125, 15624,
    78120, 390600, 1953000, 9765000, 48825000, -1<\/strong>,
    244124996, 244124976, 244124876,
    244124376, 244121876, 244109376,
    244046876, 243734376, 242171876,
    234359376, 195296876, 244109377,
    244046881, 243734401, 242172001,
    234360001, 195300001, 1 }<\/p>\n\n\n\n

    For 6103437501 from t=7 the 42 elements in NNd42<\/strong> are as shown next
    { 5, 25, 125, 625, 3125, 15625,
    78125, 390625, 1953125, 9765625, 48828125, 244140625,
    1220703125, 78124, 390620, 1953100, 9765500, 48827500,
    244137500, 1220687500, -1<\/strong>,
    6103437496, 6103437476, 6103437376,
    6103436876, 6103434376, 6103421876,
    6103359376, 6103046876, 6101484376,
    6093671876, 6054609376, 5859296876,
    4882734376, 6103359377, 6103046881,
    6101484401, 6093672001, 6054610001,
    5859300001, 4882750001, 1 }<\/p>\n\n\n\n

    For t=8 the 48 elements in NNd48<\/strong> are not tabulated here
    For t=9 the 54 elements in NNd54<\/strong> are not tabulated here
    For t=10 the 60 elements in NNd60<\/strong> are not tabulated here
    For t=11 the 66 elements in NNd66<\/strong> are not tabulated here
    For t=12 the 72 elements in NNd72<\/strong> are not tabulated here
    For t=13 the 78 elements in NNd78<\/strong> are not tabulated here
    For t=14 the 84 elements in NNd84<\/strong> are not tabulated here
    For t=15 the 90 elements in NNd90<\/strong> are not tabulated here
    For t=16 the 96 elements in NNd96<\/strong> are not tabulated here
    For t=17 the 102 elements in NNd102<\/strong> are not tabulated here
    For t=18 the 108 elements in NNd108<\/strong> are not tabulated here
    For t=19 the 114 elements in NNd114<\/strong> are not tabulated here
    For t=20 the 120 elements in NNd120<\/strong> are not tabulated here<\/p>\n\n\n\n

    Using sturdy element notation we might say the results tabulated in this post are<\/p>\n\n\n\n

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

    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 NNd to avoid clashing with existing letter conventions for groups.Prefer G or S? Replace them in your local copy. […]<\/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-251","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\/251","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=251"}],"version-history":[{"count":1,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/251\/revisions"}],"predecessor-version":[{"id":253,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/251\/revisions\/253"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=251"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=251"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}