{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:03:09Z","timestamp":1776769389948,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2003,5,16]],"date-time":"2003-05-16T00:00:00Z","timestamp":1053043200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The well-known Baillie-PSW probable prime test is a combination of a Rabin-Miller test and a \u201ctrue\u201d (i.e., with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis upper D slash n right-parenthesis equals negative 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(D\/n)=-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT) with probability of error less than 1\/7710, and pointed out that the lack of counter-examples to the Baillie-PSW test indicates that the true probability of error may be much lower.\n                  <\/p>\n                  <p>\n                    In this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter:\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T Subscript u Baseline equals upper T mod left-parenthesis upper T squared minus u upper T plus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mi>u<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mspace width=\"0.667em\"\/>\n                            <mml:mi>mod<\/mml:mi>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">T_u=T \\mod (T^2-uT+1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and define the base-counting functions:\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper B left-parenthesis n right-parenthesis equals number-sign left-brace u colon 0 less-than-or-equal-to u greater-than n comma n is a p s p left-parenthesis upper T Subscript u Baseline right-parenthesis right-brace\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">B<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              #\n                              \n                            <\/mml:mi>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"thickmathspace\"\/>\n                            <mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mtext>\u00a0is a\u00a0<\/mml:mtext>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>psp<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>T<\/mml:mi>\n                                  <mml:mi>u<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\mathrm {B}(n) = \\#\\{u: 0\\leq u&gt;n,\\; \\text {$n$ is a $\\operatorname {psp}(T_u)$}\\} \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    and\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper S normal upper B left-parenthesis n right-parenthesis equals number-sign left-brace u colon 0 less-than-or-equal-to u greater-than n comma n is an s p s p left-parenthesis upper T Subscript u Baseline right-parenthesis right-brace period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">S<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">B<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              #\n                              \n                            <\/mml:mi>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"thickmathspace\"\/>\n                            <mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mtext>\u00a0is an\u00a0<\/mml:mtext>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>spsp<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>T<\/mml:mi>\n                                  <mml:mi>u<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\mathrm {SB}(n) = \\#\\{u: 0\\leq u&gt;n,\\; \\text {$n$ is an $\\operatorname {spsp}(T_u)$}\\}. \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    Then we give explicit formulas to compute B\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and SB\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and prove that, for odd composites\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B left-parenthesis n right-parenthesis greater-than n slash 2 and SB left-parenthesis n right-parenthesis greater-than n slash 8 comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mtext>\u00a0B<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>n<\/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:mspace width=\"thickmathspace\"\/>\n                            <mml:mrow>\n                              <mml:mtext>\u00a0and<\/mml:mtext>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mspace width=\"thickmathspace\"\/>\n                              <\/mml:mrow>\n                              <mml:mtext>\u00a0SB<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>8<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*}\\text { B}(n)&gt;n\/2\\; \\text { and$\\;$ SB}(n)&gt;n\/8, \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    and point out that these are best possible. Finally, based on one-parameter quadratic-base pseudoprimes, we provide a probable prime test, called the One-Parameter Quadratic-Base Test (OPQBT), which passed by all primes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"greater-than-or-equal-to 5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\geq 5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and passed by an odd composite\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals p 1 Superscript r 1 Baseline p 2 Superscript r 2 Baseline midline-horizontal-ellipsis p Subscript s Superscript r Super Subscript s Superscript Baseline left-parenthesis p 1 greater-than p 2 greater-than midline-horizontal-ellipsis greater-than p Subscript s Baseline\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>r<\/mml:mi>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:msubsup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>r<\/mml:mi>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>r<\/mml:mi>\n                                  <mml:mi>s<\/mml:mi>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mspace width=\"thickmathspace\"\/>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=p_1^{r_1}p_2^{r_2}\\cdots p_s^{r_s}\\;(p_1&gt;p_2&gt;\\cdots &gt;p_s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    odd primes) with probability of error\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We give explicit formulas to compute\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and prove that\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis n right-parenthesis greater-than StartLayout Enlarged left-brace 1st Row 1st Column 1 slash n Superscript 4 slash 3 Baseline comma 2nd Column a m p semicolon for n nonsquare free with s equals 1 semicolon 2nd Row 1st Column 1 slash n Superscript 2 slash 3 Baseline comma 2nd Column a m p semicolon for n square free with s equals 2 semicolon 3rd Row 1st Column 1 slash n Superscript 2 slash 7 Baseline comma 2nd Column a m p semicolon for n square free with s equals 3 semicolon 4th Row 1st Column StartFraction 1 Over 8 Superscript s minus 4 Baseline dot 166 left-parenthesis p 1 plus 1 right-parenthesis EndFraction comma 2nd Column a m p semicolon for n square free with s even greater-than-or-equal-to 4 semicolon 5th Row 1st Column StartFraction 1 Over 16 Superscript s minus 5 Baseline dot 119726 EndFraction comma 2nd Column a m p semicolon for n square free with s odd greater-than-or-equal-to 5 semicolon 6th Row 1st Column StartFraction 1 Over 4 Superscript s Baseline EndFraction product Underscript i equals 1 Overscript s Endscripts StartFraction 1 Over p Subscript i Superscript 2 left-parenthesis r Super Subscript i Superscript minus 1 right-parenthesis Baseline EndFraction comma 2nd Column a m p semicolon otherwise comma i period e period comma for n nonsquare free with s greater-than-or-equal-to 2 period EndLayout\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\">\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mn>1<\/mml:mn>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mo>\/<\/mml:mo>\n                                    <\/mml:mrow>\n                                    <mml:msup>\n                                      <mml:mi>n<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>4<\/mml:mn>\n                                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                          <mml:mo>\/<\/mml:mo>\n                                        <\/mml:mrow>\n                                        <mml:mn>3<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                    <mml:mo>,<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mrow>\n                                      <mml:mtext>for\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0nonsquare free with\u00a0<\/mml:mtext>\n                                    <\/mml:mrow>\n                                    <mml:mi>s<\/mml:mi>\n                                    <mml:mo>=<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                    <mml:mo>;<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mn>1<\/mml:mn>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mo>\/<\/mml:mo>\n                                    <\/mml:mrow>\n                                    <mml:msup>\n                                      <mml:mi>n<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>2<\/mml:mn>\n                                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                          <mml:mo>\/<\/mml:mo>\n                                        <\/mml:mrow>\n                                        <mml:mn>3<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                    <mml:mo>,<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mrow>\n                                      <mml:mtext>for\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0square free with\u00a0<\/mml:mtext>\n                                    <\/mml:mrow>\n                                    <mml:mi>s<\/mml:mi>\n                                    <mml:mo>=<\/mml:mo>\n                                    <mml:mn>2<\/mml:mn>\n                                    <mml:mo>;<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mn>1<\/mml:mn>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mo>\/<\/mml:mo>\n                                    <\/mml:mrow>\n                                    <mml:msup>\n                                      <mml:mi>n<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>2<\/mml:mn>\n                                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                          <mml:mo>\/<\/mml:mo>\n                                        <\/mml:mrow>\n                                        <mml:mn>7<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                    <mml:mo>,<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mrow>\n                                      <mml:mtext>for\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0square free with\u00a0<\/mml:mtext>\n                                    <\/mml:mrow>\n                                    <mml:mi>s<\/mml:mi>\n                                    <mml:mo>=<\/mml:mo>\n                                    <mml:mn>3<\/mml:mn>\n                                    <mml:mo>;<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mfrac>\n                                      <mml:mn>1<\/mml:mn>\n                                      <mml:mrow>\n                                        <mml:msup>\n                                          <mml:mn>8<\/mml:mn>\n                                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                            <mml:mi>s<\/mml:mi>\n                                            <mml:mo>\n                                              \u2212\n                                              \n                                            <\/mml:mo>\n                                            <mml:mn>4<\/mml:mn>\n                                          <\/mml:mrow>\n                                        <\/mml:msup>\n                                        <mml:mo>\n                                          \u22c5\n                                          \n                                        <\/mml:mo>\n                                        <mml:mn>166<\/mml:mn>\n                                        <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                        <mml:msub>\n                                          <mml:mi>p<\/mml:mi>\n                                          <mml:mn>1<\/mml:mn>\n                                        <\/mml:msub>\n                                        <mml:mo>+<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                        <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                      <\/mml:mrow>\n                                    <\/mml:mfrac>\n                                    <mml:mo>,<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mrow>\n                                      <mml:mtext>for\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0square free with\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>s<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0even<\/mml:mtext>\n                                    <\/mml:mrow>\n                                    <mml:mo>\n                                      \u2265\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>4<\/mml:mn>\n                                    <mml:mo>;<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mfrac>\n                                      <mml:mn>1<\/mml:mn>\n                                      <mml:mrow>\n                                        <mml:msup>\n                                          <mml:mn>16<\/mml:mn>\n                                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                            <mml:mi>s<\/mml:mi>\n                                            <mml:mo>\n                                              \u2212\n                                              \n                                            <\/mml:mo>\n                                            <mml:mn>5<\/mml:mn>\n                                          <\/mml:mrow>\n                                        <\/mml:msup>\n                                        <mml:mo>\n                                          \u22c5\n                                          \n                                        <\/mml:mo>\n                                        <mml:mn>119726<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:mfrac>\n                                    <mml:mo>,<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mrow>\n                                      <mml:mtext>for\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0square free with\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>s<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0odd<\/mml:mtext>\n                                    <\/mml:mrow>\n                                    <mml:mo>\n                                      \u2265\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>5<\/mml:mn>\n                                    <mml:mo>;<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mfrac>\n                                      <mml:mn>1<\/mml:mn>\n                                      <mml:msup>\n                                        <mml:mn>4<\/mml:mn>\n                                        <mml:mi>s<\/mml:mi>\n                                      <\/mml:msup>\n                                    <\/mml:mfrac>\n                                    <mml:munderover>\n                                      <mml:mo>\n                                        \u220f\n                                        \n                                      <\/mml:mo>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>i<\/mml:mi>\n                                        <mml:mo>=<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>s<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:munderover>\n                                    <mml:mfrac>\n                                      <mml:mn>1<\/mml:mn>\n                                      <mml:msubsup>\n                                        <mml:mi>p<\/mml:mi>\n                                        <mml:mi>i<\/mml:mi>\n                                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                          <mml:mn>2<\/mml:mn>\n                                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                          <mml:msub>\n                                            <mml:mi>r<\/mml:mi>\n                                            <mml:mi>i<\/mml:mi>\n                                          <\/mml:msub>\n                                          <mml:mo>\n                                            \u2212\n                                            \n                                          <\/mml:mo>\n                                          <mml:mn>1<\/mml:mn>\n                                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                        <\/mml:mrow>\n                                      <\/mml:msubsup>\n                                    <\/mml:mfrac>\n                                    <mml:mo>,<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mrow>\n                                      <mml:mtext>otherwise, i.e., for\u00a0<\/mml:mtext>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mtext>\u00a0nonsquare free with\u00a0<\/mml:mtext>\n                                    <\/mml:mrow>\n                                    <mml:mi>s<\/mml:mi>\n                                    <mml:mo>\n                                      \u2265\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>2.<\/mml:mn>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                              <\/mml:mtable>\n                              <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\"\/>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\tau (n)&gt; \\begin {cases} 1\/n^{4\/3}, &amp; \\text {for $n$ nonsquare free with } s=1; \\\\ 1\/n^{2\/3}, &amp; \\text {for $n$ square free with } s=2; \\\\ 1\/n^{2\/7}, &amp; \\text {for $n$ square free with } s=3; \\\\ \\frac {1}{8^{s-4}\\cdot 166(p_1+1)}, &amp; \\text {for $n$ square free with $s$ even} \\geq 4; \\\\ \\frac {1}{16^{s-5}\\cdot 119726}, &amp; \\text {for $n$ square free with $s$ odd} \\geq 5;\\\\ \\frac {1}{4^s}\\prod _{i=1}^{s}\\frac {1}{p_i^{2(r_i-1)}}, &amp; \\text {otherwise, i.e., for $n$ nonsquare free with } s\\geq 2. \\end{cases} \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    The running time of the OPQBT is asymptotically 4 times that of a Rabin-Miller test for worst cases, but twice that of a Rabin-Miller test for most composites. We point out that the OPQBT has clear finite group (field) structure and nice symmetry, and is indeed a more general and strict version of the Baillie-PSW test. Comparisons with Gantham\u2019s RQFT are given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01424-2","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1699-1734","source":"Crossref","is-referenced-by-count":6,"title":["A one-parameter quadratic-base version of the Baillie-PSW probable prime test"],"prefix":"10.1090","volume":"71","author":[{"given":"Zhenxiang","family":"Zhang","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,5,16]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"703","DOI":"10.2307\/2118576","article-title":"There are infinitely many Carmichael numbers","volume":"139","author":"Alford, W. R.","year":"1994","journal-title":"Ann. of Math. 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Algebra 12 (1978), no. 1, 93\u2013100; MR 57 #16387]","volume":"20","author":"Wu, Yel Chiang","year":"1981","journal-title":"J. Pure Appl. 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