{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T15:25:38Z","timestamp":1694618738824},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"218","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Let <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> be a positive integer and suppose that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n  <mml:semantics>\n    <mml:mi>p<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is an odd prime with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p identical-to 1 mod m\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mo>\u2261<!-- \u2261 --><\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mo lspace=\"thickmathspace\" rspace=\"thickmathspace\">mod<\/mml:mo>\n      <mml:mi>m<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">p \\equiv 1 \\bmod m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Suppose that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a element-of left-parenthesis double-struck upper Z slash p double-struck upper Z right-parenthesis Superscript asterisk\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>a<\/mml:mi>\n      <mml:mo>\u2208<!-- \u2208 --><\/mml:mo>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:msup>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">a \\in (\\mathbb {Z}\/p\\mathbb {Z})^*<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and consider the polynomial <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x Superscript m Baseline minus a\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>x<\/mml:mi>\n        <mml:mi>m<\/mml:mi>\n      <\/mml:msup>\n      <mml:mo>\u2212<!-- \u2212 --><\/mml:mo>\n      <mml:mi>a<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">x^m-a<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. If this polynomial has any roots in <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis double-struck upper Z slash p double-struck upper Z right-parenthesis Superscript asterisk\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:msup>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">(\\mathbb {Z}\/p\\mathbb {Z})^*<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, where the coset representatives for <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z slash p double-struck upper Z\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}\/p\\mathbb {Z}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> are taken to be all integers <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u\">\n  <mml:semantics>\n    <mml:mi>u<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">u<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue u EndAbsoluteValue greater-than p slash 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>u<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi>p<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">|u|&gt;p\/2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, then these roots will form a coset of the multiplicative subgroup <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"mu Subscript m\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>\u03bc<!-- \u03bc --><\/mml:mi>\n      <mml:mi>m<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\mu _m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis double-struck upper Z slash p double-struck upper Z right-parenthesis Superscript asterisk\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:msup>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">(\\mathbb {Z}\/p\\mathbb {Z})^*<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> consisting of the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>th roots of unity mod <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n  <mml:semantics>\n    <mml:mi>p<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Let <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C\">\n  <mml:semantics>\n    <mml:mi>C<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">C<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> be a coset of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"mu Subscript m\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>\u03bc<!-- \u03bc --><\/mml:mi>\n      <mml:mi>m<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\mu _m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> in <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis double-struck upper Z slash p double-struck upper Z right-parenthesis Superscript asterisk\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n      <\/mml:mrow>\n      <mml:msup>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">(\\mathbb {Z}\/p\\mathbb {Z})^*<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, and define <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue upper C EndAbsoluteValue equals max Underscript u element-of upper C Endscripts StartAbsoluteValue u EndAbsoluteValue\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>C<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mo>=<\/mml:mo>\n      <mml:munder>\n        <mml:mo movablelimits=\"true\" form=\"prefix\">max<\/mml:mo>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>u<\/mml:mi>\n          <mml:mo>\u2208<!-- \u2208 --><\/mml:mo>\n          <mml:mi>C<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:munder>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo stretchy=\"false\">|<\/mml:mo>\n        <\/mml:mrow>\n        <mml:mi>u<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo stretchy=\"false\">|<\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:mrow>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">|C|=\\max _{u \\in C}{|u|}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. In the paper \u201cNumbers Having <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> Small <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>th Roots mod <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n  <mml:semantics>\n    <mml:mi>p<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>\u201d (<italic>Mathematics of Computation<\/italic>, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M 1 left-parenthesis m comma p right-parenthesis equals min Underscript upper C element-of left-parenthesis double-struck upper Z slash p double-struck upper Z right-parenthesis Superscript asterisk Baseline slash mu Subscript m Baseline Endscripts StartAbsoluteValue upper C EndAbsoluteValue\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>M<\/mml:mi>\n        <mml:mn>1<\/mml:mn>\n      <\/mml:msub>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>p<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>=<\/mml:mo>\n      <mml:munder>\n        <mml:mo movablelimits=\"true\" form=\"prefix\">min<\/mml:mo>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mstyle mathsize=\"0.5em\">\n            <mml:mi>C<\/mml:mi>\n            <mml:mo>\u2208<!-- \u2208 --><\/mml:mo>\n            <mml:mo stretchy=\"false\">(<\/mml:mo>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n            <\/mml:mrow>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mo>\/<\/mml:mo>\n            <\/mml:mrow>\n            <mml:mi>p<\/mml:mi>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n            <\/mml:mrow>\n            <mml:msup>\n              <mml:mo stretchy=\"false\">)<\/mml:mo>\n              <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n            <\/mml:msup>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mo>\/<\/mml:mo>\n            <\/mml:mrow>\n            <mml:msub>\n              <mml:mi>\u03bc<!-- \u03bc --><\/mml:mi>\n              <mml:mi>m<\/mml:mi>\n            <\/mml:msub>\n          <\/mml:mstyle>\n        <\/mml:mrow>\n      <\/mml:munder>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo stretchy=\"false\">|<\/mml:mo>\n        <\/mml:mrow>\n        <mml:mi>C<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo stretchy=\"false\">|<\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:mrow>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">M_1(m,p)=\\min _{\\tiny C \\in (\\mathbb {Z}\/p\\mathbb {Z})^* \/\\mu _m }{|C|}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> of the form <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M 1 left-parenthesis m comma p right-parenthesis greater-than upper K Subscript m Baseline p Superscript 1 minus 1 slash phi left-parenthesis m right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>M<\/mml:mi>\n        <mml:mn>1<\/mml:mn>\n      <\/mml:msub>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>p<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:msub>\n        <mml:mi>K<\/mml:mi>\n        <mml:mi>m<\/mml:mi>\n      <\/mml:msub>\n      <mml:msup>\n        <mml:mi>p<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mn>1<\/mml:mn>\n          <mml:mo>\u2212<!-- \u2212 --><\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n            <mml:mo>\/<\/mml:mo>\n          <\/mml:mrow>\n          <mml:mi>\u03d5<!-- \u03d5 --><\/mml:mi>\n          <mml:mo stretchy=\"false\">(<\/mml:mo>\n          <mml:mi>m<\/mml:mi>\n          <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">M_1(m,p)&gt;K_mp^{1-1\/\\phi (m)}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, where <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n  <mml:semantics>\n    <mml:mi>\u03d5<!-- \u03d5 --><\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is the Euler phi-function. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson are proven to be optimal when <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a power of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2\">\n  <mml:semantics>\n    <mml:mn>2<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> or when <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m equals 6 period\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:mn>6.<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">m=6.<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula><\/p>","DOI":"10.1090\/s0025-5718-97-00797-7","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:28Z","timestamp":1027721668000},"page":"807-822","source":"Crossref","is-referenced-by-count":2,"title":["Bounds for multiplicative cosets over fields of prime order"],"prefix":"10.1090","volume":"66","author":[{"given":"Corey","family":"Powell","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","volume-title":"Algebraic number theory","year":"1967"},{"key":"2","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4570-4","volume-title":"Measure and integral. 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