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• W2017339903 abstract "Throughout this paper <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=application/x-tex>e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes an integer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=greater-than-or-slanted-equals 3> <mml:semantics> <mml:mrow> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>geqslant 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a prime <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=identical-to 1 left-parenthesis mod e right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mspace width=thickmathspace /> <mml:mn>1</mml:mn> <mml:mtext> </mml:mtext> <mml:mspace width=0.667em /> <mml:mo stretchy=false>(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=0.333em /> <mml:mi>e</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>equiv ;1 pmod e</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. With <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals e f plus 1> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>e</mml:mi> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p = ef + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and for integers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding=application/x-tex>r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 less-than-or-slanted-equals s greater-than r less-than-or-slanted-equals e minus 1> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mi>r</mml:mi> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>e</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1 leqslant s > r leqslant e - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , certain binomial coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartBinomialOrMatrix r f Choose s f EndBinomialOrMatrix> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtable rowspacing=4pt columnspacing=1em> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>r</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>s</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>left ( {begin {array}{*{20}{c}} {rf} {sf} end {array} } right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have been determined in terms of the parameters in various binary and quaternary quadratic forms by, for example, Gauss [<bold>13</bold>], Jacobi [<bold>19</bold>, <bold>20</bold>], Stern [<bold>37</bold>-<bold>40</bold>], Lehmer [<bold>23</bold>] and Whiteman [<bold>42</bold>, <bold>45</bold>, <bold>46</bold>]. In <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=section-sign 2> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>S 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we determine for each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=application/x-tex>e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the exact number of binomial coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartBinomialOrMatrix r f Choose s f EndBinomialOrMatrix> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtable rowspacing=4pt columnspacing=1em> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>r</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>s</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>left ( {begin {array}{*{20}{c}} {rf} {sf} end {array} } right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not trivially congruent to one another by elementary properties of number theory and call these representative binomial coefficients. A representative binomial coefficient is said to be of order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=application/x-tex>e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis r comma s right-parenthesis equals 1> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>(r,s) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=section-sign section-sign 3 minus 4> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mn>3</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>S S 3-4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show how the Davenport-Hasse relation [<bold>7</bold>], in a form given by Yamamoto [<bold>50</bold>], leads to determinations of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n Superscript left-parenthesis p minus 1 right-parenthesis slash m> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{n^{(p - 1)/m}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of binomial coefficients modulo <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals e f plus 1 equals m n f plus 1> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>e</mml:mi> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p = ef + 1 = mnf + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These results are of some interest in themselves and are used extensively in later sections of the paper. Making use of Theorem 5.1 relating Jacobi sums and binomial coefficients, which was first obtained in a slightly different form by Whiteman [<bold>45</bold>], we systematically investigate in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=section-sign section-sign 6 minus 21> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mn>6</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>21</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>S S 6-21</mml:annotation> </mml:semantics> </mml:math> </inline-formula> all representative binomial coefficients of orders <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e equals 3 comma 4 comma 6 comma 7 comma 8 comma 9 comma 11 comma 12 comma 14 comma 15 comma 16 comma 20> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> <mml:mo>,</mml:mo> <mml:mn>9</mml:mn> <mml:mo>,</mml:mo> <mml:mn>11</mml:mn> <mml:mo>,</mml:mo> <mml:mn>12</mml:mn> <mml:mo>,</mml:mo> <mml:mn>14</mml:mn> <mml:mo>,</mml:mo> <mml:mn>15</mml:mn> <mml:mo>,</mml:mo> <mml:mn>16</mml:mn> <mml:mo>,</mml:mo> <mml:mn>20</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>e = 3,4,6,7,8,9,11,12,14,15,16,20</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=24> <mml:semantics> <mml:mn>24</mml:mn> <mml:annotation encoding=application/x-tex>24</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which we are able to determine explicitly in terms of the parameters in well-known binary quadratic forms, and all representative binomial coefficients of orders <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e equals 5 comma 10 comma 13 comma 15 comma 16> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>10</mml:mn> <mml:mo>,</mml:mo> <mml:mn>13</mml:mn> <mml:mo>,</mml:mo> <mml:mn>15</mml:mn> <mml:mo>,</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>e = 5,10,13,15,16</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=20> <mml:semantics> <mml:mn>20</mml:mn> <mml:annotation encoding=application/x-tex>20</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which we are able to explicitly determine in terms of quaternary quadratic decompositions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=16 p> <mml:semantics> <mml:mrow> <mml:mn>16</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>16p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by Dickson [<bold>9</bold>], Zee [<bold>51</bold>] and Guidici, Muskat and Robinson [<bold>14</bold>]. Some of these results have been obtained by previous authors and many new ones are included. For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e equals 7> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>=</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>e = 7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=14> <mml:semantics> <mml:mn>14</mml:mn> <mml:annotation encoding=application/x-tex>14</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we are unable to explicitly determine representative binomial coefficients in terms of the six variable quadratic decomposition of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=72 p> <mml:semantics> <mml:mrow> <mml:mn>72</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>72p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by Dickson [<bold>9</bold>] for reasons given in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=section-sign 10> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>S 10</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, but we are able to express these binomial coefficients in terms of the parameter <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{x_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in this system in analogy to a recent result of Rajwade [<bold>34</bold>]. Finally, although a relatively rare occurrence for small <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=application/x-tex>e</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it is possible for representative binomial coefficients of order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=application/x-tex>e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be congruent to one another <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis mod p right-parenthesis> <mml:semantics> <mml:mrow> <mml:mspace width=0.667em /> <mml:mo stretchy=false>(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=0.333em /> <mml:mi>p</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>pmod p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Representative binomial coefficients which are congruent to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=plus-or-minus 1> <mml:semantics> <mml:mrow> <mml:mo>±<!-- ± --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>pm 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> times at least one other representative for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals e f plus 1> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>e</mml:mi> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p = ef + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are called Cauchy-Whiteman type binomial coefficients for reasons given in [<bold>17</bold>] and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=section-sign 21> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mn>21</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>S 21</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. All congruences between such binomial coefficients are carefully examined and proved (with the sign ambiguity removed in each case) for all values of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=application/x-tex>e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> considered. When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e equals 24> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>=</mml:mo> <mml:mn>24</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>e = 24</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=48> <mml:semantics> <mml:mn>48</mml:mn> <mml:annotation encoding=application/x-tex>48</mml:annotation> </mml:semantics> </mml:math> </inline-formula> representative binomial coefficients, including those of lower order, and it is shown in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=section-sign 21> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>§<!-- § --></mml:mi> <mml:mn>21</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>S 21</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that an astonishing <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=43> <mml:semantics> <mml:mn>43</mml:mn> <mml:annotation encoding=application/x-tex>43</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of these are Cauchy-Whiteman type binomial coefficients. It is of particular interest that the sign ambiguity in many of these congruences does not arise from any expression of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n Superscript left-parenthesis p minus 1 right-parenthesis slash m> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{n^{(p - 1)/m}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in contrast to the case for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e greater-than 24> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>></mml:mo> <mml:mn>24</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>e > 24</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2017339903 created "2016-06-24" @default.
• W2017339903 creator A5000794153 @default.
• W2017339903 creator A5056626858 @default.
• W2017339903 date "1984-01-01" @default.
• W2017339903 modified "2023-09-25" @default.
• W2017339903 title "Binomial coefficients and Jacobi sums" @default.
• W2017339903 cites W1030561078 @default.
• W2017339903 cites W1514546851 @default.
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