Kenneth S. Williams

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All published works

Action	Title	Year	Authors
+	Extensions of an identity of Chan and Cooper in the spirit of Ramanujan	2024	Florian Münkel Lerna Pehlivan Kenneth S. Williams
+	Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions	2024	James G. Huard Zhiming M. Ou Blair K. Spearman Kenneth S. Williams
+	Explicit formulas for the Fourier coefficients of a class of eta quotients	2024	Florian Münkel Kenneth S. Williams
+	Using the Girard-Newton-Waring Formula to Evaluate Trigonometric Sums: A Blast from the Past	2024	Kenneth S. Williams
+	Infinite product representations of some q-series	2023	Florian Münkel Lerna Pehlivan Kenneth S. Williams
+	An Arithmetic Proof of a Theorem of Chan, Long, and Yang	2023	Kenneth S. Williams
+	Beyond the Basel Problem: Part II	2023	Kenneth S. Williams
+	Beyond the Basel Problem, Part I	2023	Kenneth S. Williams
+	AN APPLICATION OF BINARY QUADRATIC FORMS OF DISCRIMINANT TO MODULAR FORMS	2022	Zafer Selçuk Aygin Kenneth S. Williams
+	Representation Numbers of Spinor Regular Ternary Quadratic Forms	2021	Zafer Selçuk Aygin Greg Doyle Florian Münkel Lerna Pehlivan Kenneth S. Williams
+	Derivable quadratic forms and representation numbers	2020	Zafer Selçuk Aygin Kenneth S. Williams
+	Why Does a Prime p Divide a Fermat Number?	2020	Zafer Selçuk Aygin Kenneth S. Williams
+	Invitation To Algebra: A Resource Compendium For Teachers, Advanced Undergraduate Students And Graduate Students In Mathematics	2020	Kenneth S. Williams Vlastimil Dlab
+	Representation numbers of seven quaternary quadratic forms each in a genus consisting of only two classes	2020	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Invitation to Algebra	2020	Vlastimil Dlab Kenneth S. Williams
+	PRIME-UNIVERSAL QUADRATIC FORMS AND	2019	Greg Doyle Kenneth S. Williams
+	Positive Integers Represented by Regular Primitive Positive-Definite Integral Ternary Quadratic Forms	2019	Gregory Doyle Joseph B. Muskat Lerna Pehlivan Kenneth S. Williams
+	The Many Sides of the Pythagorean Theorem	2019	Vlastimil Dlab Kenneth S. Williams
+	Some Formulas of Liouville in the Spirit of Gauss	2019	Kenneth S. Williams
+	Everything You Wanted To Know About <i>ax</i><sup>2</sup>+<i>by</i><sup>2</sup>+<i>cz</i><sup>2</sup>+<i>dt</i><sup>2</sup> But Were Afraid To Ask	2018	Kenneth S. Williams
+	Evaluation of some q-integrals in terms of the Dedekind eta function	2018	Greg Doyle Kenneth S. Williams
+	Positive-Definite Ternary Quadratic Forms Which are (4, 1)-universal and (4, 3)-universal.	2018	Lerna Pehlivan Kenneth S. Williams
+	(k, l)-Universality of Ternary Quadratic Forms ax2+by2+cz2.	2018	Lerna Pehlivan Kenneth S. Williams
+	A Positive-Definite Ternary Quadratic Form Does Not Represent All Positive Integers.	2017	Greg Doyle Kenneth S. Williams
+	Infinite products with coefficients which vanish on certain arithmetic progressions	2016	Ayşe Alaca Şaban Alaca Zafer Selçuk Aygin Kenneth S. Williams
+	Bounds for the Representations of Integers by Positive Quadratic Forms	2016	Kenneth S. Williams
+	Some arithmetic convolution identities	2016	Kenneth S. Williams
+	A class of subsums of Euler’s sum	2016	Kenneth S. Williams
+	Representation numbers of certain quaternary quadratic forms in a genus consisting of a single class	2015	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Quartic residuacity and the quadratic character of certain quadratic irrationalities	2015	Ronald J. Evans Kenneth S. Williams
+ PDF Chat	Ternary Quadratic Forms and Eta Quotients	2015	Kenneth S. Williams
+	Advances in the Theory of Numbers : Proceedings of the Thirteenth Conference of the Canadian Number Theory Association	2015	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	A “Four Integers” Theorem and a “Five Integers” Theorem	2015	Kenneth S. Williams
+	Historical Remark on Ramanujan′s Tau Function	2015	Kenneth S. Williams
+	Advances in the Theory of Numbers	2015	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+ PDF Chat	Some new evaluations of the Legendre symbol \bigl( \fraca+b\sqrtqp \bigr)	2015	Lerna Pehlivan Kenneth S. Williams
+	Historical remark on a theorem of Zhang and Yue	2014	Kenneth S. Williams
+	On the number of representations of a positive integer as a sum of two binary quadratic forms	2014	Şaban Alaca Lerna Pehlivan Kenneth S. Williams
+	Analogues of Ramanujan's 24 squares formula	2014	Faruk Uygul Kenneth S. Williams
+ PDF Chat	Ternary quadratic forms ax<sup>2</sup>+by<sup>2</sup>+cz<sup>2</sup>representing all positive integers 8k+4	2014	Kenneth S. Williams
+	The power series expansion of certain infinite products q r ∏ n = 1 ∞ ( 1 − q n ) a 1 ( 1 − q 2 n ) a 2 ⋯ ( 1 − q m n ) a m $q^{r}\prod_{n=1}^{\infty}(1-q^{n})^{a_{1}}(1-q^{2n})^{a_{2}}\cdots(1-q^{mn})^{a_{m}}$	2013	Lerna Pehlivan Kenneth S. Williams
+	EULER PRODUCTS IN RAMANUJAN'S LOST NOTEBOOK	2013	Bruce C. Berndt Byungchan Kim Kenneth S. Williams
+ PDF Chat	Quadratic forms and a product-to-sum formula	2013	Kenneth S. Williams
+	The Parents of Jacobi’s Four Squares Theorem Are Unique	2013	Kenneth S. Williams
+ PDF Chat	Some arithmetic identities involving divisor functions	2012	Şaban Alaca Faruk Uygul Kenneth S. Williams
+	ON THE QUATERNARY FORMS x<sup>2</sup>+y<sup>2</sup>+2z<sup>2</sup>+3t<sup>2</sup>, x<sup>2</sup>+2y<sup>2</sup>+2z<sup>2</sup>+6t<sup>2</sup>, x<sup>2</sup>+3y<sup>2</sup>+3z<sup>2</sup>+6t<sup>2</sup> AND 2x<sup>2</sup>+3y<sup>2</sup>+6z<sup>2</sup>+6t<sup>2</sup>	2012	Ayşe Alaca Kenneth S. Williams
+	FOURIER SERIES OF A CLASS OF ETA QUOTIENTS	2012	Kenneth S. Williams
+	Maximal unbalanced families	2012	Louis J. Billera Justin Tatch Moore C. Dufort Moraites Y. Wang Kenneth S. Williams
+	ON THE QUATERNARY FORMS x 2 + y 2 +2 z 2 +3 t 2 , x 2 +2 y 2 +2 z 2 +6 t 2 , x 2 +3 y 2 +3 z 2 +6 t 2 AND 2x 2 +3 y 2 +6 z 2 +6 t 2	2012	Ayşe Alaca Kenneth S. Williams
+	SOME PRODUCT-TO-SUM IDENTITIES	2012	Şaban Alaca Lerna Pehlivan Kenneth S. Williams
+ PDF Chat	Lifting monogenic cubic fields to monogenic sextic fields	2011	Melisa J. Lavallee Blair K. Spearman Kenneth S. Williams
+	RESTRICTED EISENSTEIN SERIES AND CERTAIN CONVOLUTION SUMS	2011	Ayşe Alaca Faruk Uygul Kenneth S. Williams
+	Some Twisted Convolution Sums	2010	Kenneth S. Williams
+	Jacobi's Four Squares Formula	2010	Kenneth S. Williams
+	A Recurrence Relation for σ*(<i>n</i>)	2010	Kenneth S. Williams
+	Sums of Two, Four, Six and Eight Triangular Numbers	2010	Kenneth S. Williams
+	Sums of integers of the form <i>x</i><sup>2</sup> + <i>xy</i> + <i>y</i><sup>2</sup>	2010	Kenneth S. Williams
+	The Girard-Fermat Theorem	2010	Kenneth S. Williams
+	Four Elementary Arithmetic Formulae	2010	Kenneth S. Williams
+	The Arithmetic Functions σ<sub> <i>k</i> </sub>(<i>n</i>), σ<sub> <i>k</i> </sub>*(<i>n</i>), <i>d<sub>k, m</sub>(n)</i> and <i>F<sub>k</sub>(n)</i>	2010	Kenneth S. Williams
+	Sums of Eight and Twelve Squares	2010	Kenneth S. Williams
+	Number Theory in the Spirit of Liouville	2010	Kenneth S. Williams
+	Liouville's Ideas in Number Theory	2010	Kenneth S. Williams
+ PDF Chat	The number of representations of a positive integer by certain octonary quadratic forms	2010	Şaban Alaca Kenneth S. Williams
+	SEXTENARY QUADRATIC FORMS AND AN IDENTITY OF KLEIN AND FRICKE	2010	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	FOURTEEN OCTONARY QUADRATIC FORMS	2010	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+ PDF Chat	Representations by sextenary quadratic forms whose coefficients are 1, 2 and 4	2010	Ayşe Alaca Şaban Alaca Faruk Uygul Kenneth S. Williams
+ PDF Chat	Some Infinite Products of Ramanujan Type	2009	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Evaluation of the sums % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaaabCaeaacqaHdpWCcqGGOaakcqWGTbqBcqGG % PaqkcqaHdpWCcqGGOaakcqWGUbGBcqGHsislcqWGTbqBcqGGPaqkaS % …	2009	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	THE NUMBER OF REPRESENTATIONS OF A POSITIVE INTEGER BY CERTAIN QUATERNARY QUADRATIC FORMS	2009	Ayşe Alaca Şaban Alaca Mathieu F. Lemire Kenneth S. Williams
+	SUMS OF 4k SQUARES: A POLYNOMIAL APPROACH	2009	Kenneth S. Williams
+	SOME NEW THETA FUNCTION IDENTITIES WITH APPLICATIONS TO SEXTENARY QUADRATIC FORMS	2009	Ayşe Alaca Kenneth S. Williams
+	Some identities involving theta functions	2008	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Liouville's sextenary quadratic forms x2 + y2 + z2 + t2 + 2u2 + 2v2, x2 + y2 + 2z2 + 2t2 + 2u2 + 2v2 and x2 + 2y2 + 2z2 + 2t2 + 2u2 + 4v2	2008	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	BERNDT'S CURIOUS FORMULA	2008	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	THETA FUNCTION IDENTITIES AND REPRESENTATIONS BY CERTAIN QUATERNARY QUADRATIC FORMS	2008	Ayşe Alaca Şaban Alaca Mathieu F. Lemire Kenneth S. Williams
+	The Convolution Sum Σ<sub><i>m</i><<i>n</i>/16</sub><i>σ</i>(<i>m</i>)<i>σ</i>(<i>n</i> – 16<i>m</i>)	2008	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Arithmetic Progressions and Binary Quadratic Forms	2008	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+ PDF Chat	Seven octonary quadratic forms	2008	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Indices of Integers in Cyclic Cubic Fields	2008	Blair K. Spearman Kenneth S. Williams
+	On the Representations of a Positive Integer by the Forms x 2 + y 2 + z 2 + 2t 2 and x 2 + 2y 2 + 2z 2 + 2t 2	2008	Kenneth S. Williams
+	A Normal Relative Integral Basis for the Normal Closure of a Pure Cubic Field over Q( √ −3)	2008	Blair K. Spearman Kenneth S. Williams
+	$A_4$-Sextic Fields with a Power Basis	2007	Daniël Eloff Blair K. Spearman Kenneth S. Williams
+	THE CONDUCTOR OF LECACHEUX'S PARAMETRIC FAMILY OF CYCLIC QUINTIC FIELDS	2007	Alan K. Silvester Blair K. Spear Kenneth S. Williams
+	A Number Field with Infinitely Many Normal Integral Bases	2007	Daniël Eloff Blair K. Spearman Kenneth S. Williams
+ PDF Chat	An infinite class of identities	2007	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Reducibility and the Galois Group of a Parametric Family of Quintic Polynomials	2007	Melisa J. Lavallee Blair K. Spearman Kenneth S. Williams
+	Evaluation of the Convolution Sums ∑ l+24m=n σ(l) σ(m) and ∑3l+8m=n σ(l) σ(m)	2007	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+ PDF Chat	On the Common Index Divisors of a Dihedral Field of Prime Degree	2007	Blair K. Spearman Kenneth S. Williams Qiduan Yang
+ PDF Chat	Nineteen quaternary quadratic forms	2007	Ayşe Alaca Şaban Alaca Mathieu F. Lemire Kenneth S. Williams
+	THE SIMPLEST PROOF OF JACOBI'S SIX SQUARES THEOREM	2007	Kenneth S. Williams
+	Evaluation of the convolution sums	2007	Ayşe Alaca Kenneth S. Williams
+	Jacobi's Identity and Representations of Integers by Certain	2007	Ayşe Alaca Mathieu F. Lemire Kenneth S. Williams
+	Evaluation of the convolution sums \sum_{l+18m=n} \sigma(l) \sigma(m) and \sum_{2l+9m=n} \sigma(l) \sigma(m)	2007	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	Evaluation of the convolution sums <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>+</mml:mo><mml:mn>6</mml:mn><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false…	2006	Şaban Alaca Kenneth S. Williams
+ PDF Chat	EVALUATION OF COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND AT SINGULAR MODULI	2006	Habib Muzaffar Kenneth S. Williams
+ PDF Chat	The convolution sum ∑<sub>m<n⁄8</sub>σ(m) σ(n−8m)	2006	Kenneth S. Williams
+ PDF Chat	On a Double Series of Chan and Ong	2006	Kenneth S. Williams
+ PDF Chat	Cyclic Cubic Fields of Given Conductor and Given Index	2006	Alan K. Silvester Blair K. Spearman Kenneth S. Williams
+	ON THE DISTRIBUTION OF CYCLIC CUBIC FIELDS WITH INDEX 2	2006	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Values of the Euler phi function not divisible by a given odd prime	2006	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	PSL(2, 5) sextic fields with a power basis	2006	Blair K. Spearman Aya Watanabe Kenneth S. Williams
+ PDF Chat	Evaluation of two convolution sums involving the sum of divisors function	2006	Mathieu F. Lemire Kenneth S. Williams
+ PDF Chat	On the number of representations of n by ax<sup>2</sup>+bxy+cy<sup>2</sup>	2006	Zhi-Hong Sun Kenneth S. Williams
+ PDF Chat	Explicit decomposition of a rational prime in a cubic field	2006	Şaban Alaca Blair K. Spearman Kenneth S. Williams
+ PDF Chat	On the two-dimensional theta functions of the Borweins	2006	Ayşe Alaca Şaban Alaca Kenneth S. Williams
+	The Prime Ideal Factorization of 2 in Pure Quartic Fields with Index 2	2006	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Ramanujan identities and Euler products for a type of Dirichlet series	2006	Zhi-Hong Sun Kenneth S. Williams
+ PDF Chat	An arithmetic formula of Liouville	2006	Erin McAfee Kenneth S. Williams
+ PDF Chat	Integral bases for an infinite family of cyclic quintic fields	2006	Daniël Eloff Blair K. Spearman Kenneth S. Williams
+	Irreducible Quartic Polynomials with Factorizations modulo<i>p</i>	2005	Eric Driver Philip Leonard Kenneth S. Williams
+ PDF Chat	Irreducible Quartic Polynomials with Factorizations modulo p	2005	Eric Driver Philip Leonard Kenneth S. Williams
+	THE CONVOLUTION SUM $\sum\limits_{m<n/9}\sigma(m)\sigma(n-9m)$	2005	Kenneth S. Williams
+ PDF Chat	Sums of Sixteen and Twenty-Four Triangular Numbers	2005	James G. Huard Kenneth S. Williams
+	DIHEDRAL QUINTIC FIELDS WITH A POWER BASIS	2005	Melissa J. Lavallee Blair K. Spearman Kenneth S. Williams Qiduan Yang
+	WATSON'S METHOD OF SOLVING A QUINTIC EQUATION	2005	Melisa J. Lavallee Blair K. Spearman Kenneth S. Williams
+	Infinitely Many Insolvable Diophantine Equations	2004	Noriaki Kimura Kenneth S. Williams
+	Density of integers which are discriminants of cyclic fields of odd prime degree	2004	Blair K. Spearman Kenneth S. Williams
+	Infinitely Many Insolvable Diophantine Equations	2004	Noriaki Kimura Kenneth S. Williams
+	A cubic transformation formula for $_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)$ and some arithmetic convolution formulae	2004	Kenneth S. Williams
+ PDF Chat	CONVOLUTION SUMS INVOLVING THE DIVISOR FUNCTION	2004	Nathalie Cheng Kenneth S. Williams
+ PDF Chat	The proportion of cyclic quartic fields with discriminant divisible by a given prime	2004	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	On the number of representations of a positive integer by a binary quadratic form	2004	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics	2004	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	On the Lagrange resolvents of a dihedral quintic polynomial	2004	Blair K. Spearman Kenneth S. Williams
+	Applications to Diophantine Equations	2003	Şaban Alaca Kenneth S. Williams
+ PDF Chat	Units in Real Quadratic Fields	2003	Şaban Alaca Kenneth S. Williams
+	Algebraic Extensions of a Field	2003	Şaban Alaca Kenneth S. Williams
+	Factoring Primes in a Number Field	2003	Şaban Alaca Kenneth S. Williams
+	Algebraic Number Fields	2003	Şaban Alaca Kenneth S. Williams
+	Introductory Algebraic Number Theory	2003	Şaban Alaca Kenneth S. Williams
+	The Index of a Cyclic Quartic Field	2003	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	The Discriminant of a Cyclic Field of Odd Prime Degree	2003	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	The 2-Power Degree Subfields of the Splitting Fields of Polynomials with Frobenius Galois Groups	2003	Blair K. Spearman Kenneth S. Williams Qiduan Yang
+ PDF Chat	A relative integral basis over ℚ(−3) for the normal closure of a pure cubic field	2003	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Sums of twelve squares	2003	James G. Huard Kenneth S. Williams
+ PDF Chat	Commentary on an unpublished lecture by G. N. Watson on solving the quintic	2002	Bruce C. Berndt Blair K. Spearman Kenneth S. Williams
+ PDF Chat	The Discriminant of a Dihedral Quintic Field Defined by a Trinomial <i>X</i><sup>5</sup> + <i>aX</i> + <i>b</i>	2002	Blair K. Spearman Kenneth S. Williams
+	Cubic Fields with Index 2	2002	Blair K. Spearman Kenneth S. Williams
+	Uniquely Determined Unknowns in Systems of Linear Equations	2002	Kenneth Hardy Kenneth S. Williams Blair K. Spearman
+ PDF Chat	The genera representing a positive integer	2002	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	A restricted Epstein zeta function and the evaluation of some definite integrals	2002	Habib Muzaffar Kenneth S. Williams
+	A restricted Epstein zeta function and the evaluation of some denite integrals	2002	Habib Muzaffar Kenneth S. Williams
+	THE CUBIC CONGRUENCE <i>x</i><sup>3</sup> + <i>Ax</i><sup>2</sup> + <i>Bx</i> + <i>C</i> ≡ 0 (mod <i>p</i>) AND BINARY QUADRATIC FORMS II	2001	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Cubic Fields with a Power Basis	2001	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum	2001	Alfred J. van der Poorten Kenneth S. Williams
+ PDF Chat	On the Density of Cyclic Quartic Fields	2001	Zhiming M. Ou Kenneth S. Williams
+	AN ARITHMETIC PROOF OF JACOBI'S EIGHT SQUARES THEOREM	2001	Kenneth S. Williams
+ PDF Chat	Pure Quintic Fields Defined by Trinomials	2000	Blair K. Spearman Kenneth S. Williams
+	Class Number Congruences	2000	Jerzy Urbanowicz Kenneth S. Williams
+	Congruences Between the Orders of K2-Groups	2000	Jerzy Urbanowicz Kenneth S. Williams
+	Congruences among the Values of 2-Adic L-Functions	2000	Jerzy Urbanowicz Kenneth S. Williams
+	Applications of Zagier’s Formula (I)	2000	Jerzy Urbanowicz Kenneth S. Williams
+	Applications of Zagier’s Formula (II)	2000	Jerzy Urbanowicz Kenneth S. Williams
+	Congruences for L-Functions	2000	Jerzy Urbanowicz Kenneth S. Williams
+	Gauss’ congruence from Dirichlet’s class number formula and generalizations	1999	Glenn J. Fox Jerzy Urbanowicz Kenneth S. Williams
+	Dihedral quintic polynomials and a theorem of Galois	1999	K B K Blair Kenneth S. Williams
+	Number Theory: Fifth Conference of the Canadian Number Theory Association August 17-22, 1996 Carleton University, Ottawa, Ontario, Canada	1999	Rajiv Gupta Kenneth S. Williams
+ PDF Chat	Values of the Dedekind Eta Function at Quadratic Irrationalities	1999	Alfred van der Poorten Kenneth S. Williams
+	None	1999	Kenneth S. Williams
+	DEMOIVRE'S QUINTIC AND A THEOREM OF GALOIS	1999	Blair K. Spearman Kenneth S. Williams
+	Quartic Extensions of &#8474	1998	Kenneth S. Williams Blair K. Spearman Patrick J. Morandi
+	Gauss and Jacobi sums	1998	Bruce C. Berndt Kenneth S. Williams Ronald J. Evans
+	Pascal's Triangle (mod 8)	1998	James G. Huard Blair K. Spearman Kenneth S. Williams
+ PDF Chat	The conductor of a cyclic quartic field using Gauss sums	1997	Blair K. Spearman Kenneth S. Williams
+	Bernoulli's Identity Without Calculus	1997	Kenneth S. Williams
+	Bernoulli's Identity without Calculus	1997	Kenneth S. Williams
+	QUADRATIC SUBFIELDS OF THE SPLITTING FIELD OF A DIHEDRAL QUINTIC TRINOMIAL x5+ax+b	1997	Blair K. Spearman Laura Y. Spearman Kenneth S. Williams
+ PDF Chat	On Pascal's triangle modulo $p^2$	1997	James G. Huard Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Pascal's triangle (mod 9)	1997	James G. Huard Blair K. Spearman Kenneth S. Williams
+	Discriminant of the normal closure of a dihedral quartic field	1996	James G. Huard Blair K. Spearman Kenneth S. Williams
+	Relative integral bases for quartic fields over quadratic subfields	1996	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	On Solvable Quintics $X^5+aX+b$ and $X^5+aX^2+b$	1996	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	ON THE EPSTEIN ZETA FUNCTION	1996	Zhang Nan-Yue Kenneth S. Williams
+ PDF Chat	On Tornheim's double series	1996	James G. Huard Kenneth S. Williams Zhang Nan-Yue
+ PDF Chat	Normal relative integral bases for quartic fields over quadratic subfields	1996	Blair K. Spearman Kenneth S. Williams
+	The conductor of a cyclic quartic field	1996	Blair K. Spearman Kenneth S. Williams
+	The Intersection of Two Cyclotomic Extensions of a Quadratic Field	1996	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	An Arithmetic Approach to the Davenport-Hasse Relation over $GF(p)$	1995	James G. Huard Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Unramified Quadratic Extensions of a Quadratic Field	1995	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Values of the Riemann zeta function and integrals involving log(2 sinh(theta/2) and log(2 sin(theta/2)	1995	Zhang Nan-Yue Kenneth S. Williams
+	Integral Bases for Quartic Fields with Quadratic Subfields	1995	James G. Huard Blair K. Spearman Kenneth S. Williams
+ PDF Chat	The Chowla-Selberg formula for genera	1995	James G. Huard Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Predictive criteria for the representation of primes by binary quadratic forms	1995	Joseph B. Muskat Blair K. Spearman Kenneth S. Williams
+ PDF Chat	The Primes for Which an Abelian Cubic Polynomial Splits	1994	James G. Huard Blair K. Spearman Kenneth S. Williams
+	Characterization of Solvable Quintics <i>x</i><sup>5</sup> + <i>ax</i> + <i>b</i>	1994	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	REPRESENTATION OF PRIMES BY THE PRINCIPAL FORM OF NEGATIVE DISCRIMINANT $\Delta$ WHEN $h(\Delta)$ IS 4	1994	Kenneth S. Williams D. LIU
+	Characterization of Solvable Quintics x 5 + ax + b	1994	Blair K. Spearman Kenneth S. Williams
+	Comparison of the Lengths of the Continued Fractions of √D and ½(1 + √D)	1994	Kenneth S. Williams Nicholas Buck
+ PDF Chat	Comparison of the lengths of the continued fractions of √𝐷 and \frac12(1+√𝐷)	1994	Kenneth S. Williams Nicholas Buck
+	An elementary remark on the distribution of integers representable by a positive-definite integral binary quadratic form	1994	Pierre Kaplan Kenneth S. Williams
+	A short proof of the formula for the conductor of an abelian cubic field	1994	James G. Huard Blair K. Spearman Kenneth S. Williams
+	Evaluation of two trigonometric sums	1994	Kenneth S. Williams Nan Yue Zhang
+ PDF Chat	Some Series Representations of $\z(2n+1)$	1993	Zhang Yue Kenneth S. Williams
+	Representation of Primes in Arithmetic Progression by Binary Quadratic Forms	1993	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Application of the Hurwitz Zeta Function to the Evaluation of Certain Integrals	1993	Zhang Yue Kenneth S. Williams
+ PDF Chat	Special Values of the Lerch Zeta Function and the Evaluation of Certain Integrals	1993	Kenneth S. Williams Zhang Nan-Yue
+ PDF Chat	The squareroot of an ambiguous form in the principal genus	1993	Kenneth Hardy Kenneth S. Williams
+ PDF Chat	Special values of the Lerch zeta function and the evaluation of certain integrals	1993	Kenneth S. Williams Nan Yue Zhang
+	The Cubic Congruence <i>x</i> <sub>3</sub> + <i>Ax</i> <sub>2</sub> + <i>Bx</i> + C ≡ 0(mod <i>p</i> ) and Binary Quadratic Forms	1992	Blair K. Spearman Kenneth S. Williams
+	Representing Primes by Binary Quadratic Forms	1992	Blair K. Spearman Kenneth S. Williams
+	Representing Primes by Binary Quadratic Forms	1992	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	An upper bound for the sum ∑^{𝑎+𝐻}_{𝑛=𝑎+1}𝑓(𝑛) for a certain class of functions 𝑓	1992	Edward Dobrowolski Kenneth S. Williams
+	An Upper Bound for the Sum ∑ a + H n = a + 1 f(n) for a Certain Class of Functions f	1992	Edward Dobrowolski Kenneth S. Williams
+ PDF Chat	Determination of $∑_{k=1}^{p-1} g^{ak²}$ modulo p	1992	Kenneth S. Williams Kenneth Hardy
+	On an assertion of Ramanujan concerning binary quadratic forms	1991	Kenneth S. Williams
+ PDF Chat	Representation of primes by the principal form of discriminant -D when the classnumber h(-D) is 3	1991	Kenneth S. Williams Richard H. Hudson
+	Explicit evaluation of certain Eisenstein sums	1990	Kenneth S. Williams Kenneth Hardy Blair K. Spearman
+	Evaluation of the Infinite Series ∑ ∞ \substackn = 1( n p ) = 1 � � n p � � 4   n -1	1990	Kenneth Hardy Kenneth S. Williams
+	A Deterministic Algorithm for Solving n = fu 2 + gυ 2 in Coprime Integers u and υ	1990	K. J. Hardy Joseph B. Muskat Kenneth S. Williams
+ PDF Chat	The integers of a cyclic quartic field	1990	Robert H. E. Hudson Kenneth S. Williams
+ PDF Chat	Evaluation of the infinite series ∑^{∞}_{𝑛=1,(\frac𝑛𝑝)=1}(\frac𝑛𝑝)₄𝑛⁻¹	1990	Kenneth Hardy Kenneth S. Williams
+ PDF Chat	On Eisenstein's problem	1990	Noburo Ishii Pierre Kaplan Kenneth S. Williams
+ PDF Chat	A deterministic algorithm for solving $n=fu\sp 2+gv\sp 2$ in coprime integers $u$ and $v$	1990	Kenneth Hardy Joseph B. Muskat Kenneth S. Williams
+ PDF Chat	The representation of a pair of integers by a pair of positive-definite binary quadratic forms	1989	Kenneth Hardy Pierre Kaplan Kenneth S. Williams
+ PDF Chat	The Representation of a Pair of Integers by a Pair of Positive-Definite Binary Quadratic Forms	1989	K. J. Hardy Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Determination of all imaginary cyclic quartic fields with class number 2	1989	Kenneth Hardy Richard H. Hudson David R. Richman Kenneth S. Williams
+	Determination of all Imaginary Cyclic Quartic Fields with Class Number 2	1989	K. J. Hardy Richard H. Hudson David B. Richman Kenneth S. Williams
+ PDF Chat	Cyclic Quartic Fields with Relative Integral Bases Over Their Quadratic Subfields	1988	Blair K. Spearman Kenneth S. Williams
+	The Red Book: 100 Practice Problems for Undergraduate Mathematics Competitions	1988	Kenneth S. Williams Kenneth V. Hardy
+ PDF Chat	Cyclic quartic fields with relative integral bases over their quadratic subfields	1988	Blair K. Spearman Kenneth S. Williams
+	Congruences modulo 16 for the class numbers of complex quadratic fields	1987	Kenneth Hardy Kenneth S. Williams
+ PDF Chat	Calculation of the Class Numbers of Imaginary Cyclic Quartic Fields	1987	Kenneth Hardy Robert H. E. Hudson David R. Richman Kenneth S. Williams N. M. Holtz
+ PDF Chat	The cyclotomic numbers of order fifteen	1987	Nicholas Buck Lones Smith Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Calculation of the class numbers of imaginary cyclic quartic fields	1987	Kenneth Hardy Robert H. E. Hudson David R. Richman Kenneth S. Williams N. M. Holtz
+	Cyclotomy of Order Twelve Over GF(p 2 ), p 2 ≡1 (mod 12).	1987	Joseph B. Muskat Kenneth S. Williams
+ PDF Chat	The Cyclotomic Numbers of Order Fifteen	1987	Nicholas Buck Lones Smith Blair K. Spearman Kenneth S. Williams
+	THE CLASS NUMBER TWO PROBLEM FOR CERTAIN QUARTIC FIELDS(Algebraic Number Theory)	1987	Kenneth S. Williams
+	Table of the relative class numbers h[](K) of imaginary cyclic quartic fields K with h[](K)=2(MOD4) and conductor f < 416,000	1987	Kenneth Hardy Robert H. E. Hudson David R. Richman Kenneth S. Williams
+ PDF Chat	On the solvability of the Diophantine equation<i>dV</i><sup>2</sup>− 2<i>eV W</i>−<i>dW</i><sup>2</sup>= 1	1986	Kenneth Hardy Kenneth S. Williams
+	Divisibilité par 16 du nombre des classes au sens strict des corps quadratiques réels dont le deux-groupe des classes est cyclique	1986	Pierre Kaplan Kenneth S. Williams Kenneth Hardy
+	Pell's equations X2 − mY2 = −1, −4 and continued fractions	1986	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Another proof of Eisenstein's law of cubic receprocity and its supplement	1986	Christian Friesen Blair K. Spearman Kenneth S. Williams
+ PDF Chat	On the divisibility of the class number of $Q(\sqrt{pq})$ by -16	1985	Philip Leonard Kenneth S. Williams
+ PDF Chat	Criteria for Biquadratic Residuacity Modulo a Prime <i>p</i> involving Quaternary Representations of <i>p</i>	1985	Kenneth S. Williams Christian Friesen Lawrence J. Howe
+	Remark on the Class Number of Q(√2p) Modulo 8 for p ≡5 (mod 8) a Prime	1985	Kenneth S. Williams Christian Friesen
+ PDF Chat	Remark on the class number of 𝑄(√2𝑝) modulo 8 for 𝑝≡5(𝑚𝑜𝑑8) a prime	1985	Kenneth S. Williams Christian Friesen
+	Extension of a theorem of Cauchy and Jacobi	1984	Duncan A. Buell Richard H. Hudson Kenneth S. Williams
+	On the strict class number of Q(2p - - √) modulo 16, p≡1 (mod8) prime	1984	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Binomial Coefficients and Jacobi Sums	1984	Richard H. Hudson Kenneth S. Williams
+ PDF Chat	Binomial coefficients and Jacobi sums	1984	Richard H. Hudson Kenneth S. Williams
+	Table of primitive roots of primes less than 50,000	1984	Kenneth Hardy Kenneth S. Williams
+	Table of solutions of the Diophantine system : [mathematical equation given] : for primes p ≡ 1(MOD 7), p ≦ 49,939	1984	Kenneth Hardy Kenneth S. Williams
+ PDF Chat	An application of dihedral fields to representations of primes by binary quadratic forms	1984	Pierre Kaplan Kenneth S. Williams Yoshihiko Yamamoto
+	An observation on binary quadratic forms of discriminant - 32q	1983	Philip Leonard Kenneth S. Williams
+ PDF Chat	On the divisibility of the class number of by 16	1983	Philip Leonard Kenneth S. Williams
+ PDF Chat	On the Quadratic Residues (Mod<i>p</i>) in the Interval (0,<i>p</i>/4)	1983	Kenneth S. Williams
+	On Legendre's equation ax2 + by2 + cz2 = 0	1983	Richard H. Hudson Kenneth S. Williams
+ PDF Chat	Extensions of theorems of Cunningham-Aigner and Hasse-Evans	1983	Richard H. Hudson Kenneth S. Williams
+	ON THE DIVISIBILITY OF THE CLASS NUMBER OF Q(~T) BY 16	1983	Philip Leonard Kenneth S. Williams
+	Table of the 2 : strict class group and principal factors of the real quadratic fields whose strict class numbers are divisible by 16	1983	Christian Friesen Pierre Kaplan Kenneth S. Williams
+	An Artin character and representations of primes by binary quadratic forms II	1982	Franz Halter‐Koch Pierre Kaplan Kenneth S. Williams
+	Congruences modulo 8 for the class numbers of Q(√±p), p ≡ 3(mod 4) a prime	1982	Kenneth S. Williams
+ PDF Chat	Class Numbers and Biquadratic Reciprocity	1982	Kenneth S. Williams James D. Currie
+ PDF Chat	On the Divisibility of the Class Numbers of <i>Q</i>(√−<i>p</i>) and <i>Q</i>(√−2<i>p</i>) by 16.	1982	Philip Leonard Kenneth S. Williams
+ PDF Chat	Cauchy-Type Congruences for Binomial Coefficients	1982	Richard H. Hudson Kenneth S. Williams
+ PDF Chat	Resolution of ambiguities in the evaluation of cubic and quartic Jacobsthal sums	1982	Richard Hudson Kenneth S. Williams
+ PDF Chat	Class number formulae of Dirichlet type	1982	Richard H. Hudson Kenneth S. Williams
+ PDF Chat	Cauchy-type congruences for binomial coefficients	1982	Richard H. Hudson Kenneth S. Williams
+	Cauchy-typec ongruencesf or binomialc oefficients	1982	H R H Richard Kenneth S. Williams
+ PDF Chat	On the class numbers of Q(√±2p) modulo 16, for p ≡ 1 (mod 8) a prime	1982	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Congruences modulo 16 for the class numbers of the quadratic fields Q(√±p) and Q(√±2p) for p a prime congruent to 5 modulo 8	1982	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	Congruences for representations of primes by binary quadratic forms	1982	Richard H. Hudson Kenneth S. Williams
+	A representation problem involving binary quadratic forms	1981	Philip Leonard Kenneth S. Williams
+ PDF Chat	A new formulation of the law of octic reciprocity for primes<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$ \equiv \pm 3\left( {\bmod 8} \right)$" id="E1"><mml:mrow><mml:mo>≡</mml:mo><mml:mo>±</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>mod</mml:mo><mml:mn>8</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>and its consequences	1981	Richard H. Hudson Kenneth S. Williams
+	An Artin character and representations of primes by binary quadratic forms	1981	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	The class number of $Q(\sqrt{-2p}) modulo 8, for $p\equiv5 \mod 8$ is prime	1981	Kenneth S. Williams
+ PDF Chat	On the class number Q(√-p) modulo 16, for p ≡ 1 (mod 8) a prime	1981	Kenneth S. Williams
+ PDF Chat	A divisibility property of binomial coefficients viewed as an elementary sieve	1981	Richard H. Hudson Kenneth S. Williams
+ PDF Chat	Note on the quadratic character of a quadratic unit	1981	Duncan A. Buell Philip Leonard Kenneth S. Williams
+	A class number formula for certain quartic fields	1981	Richard H. Hudson Kenneth S. Williams
+ PDF Chat	Some new residuacity criteria	1980	Richard H. Hudson Kenneth S. Williams
+ PDF Chat	On the evaluation of $(q_1q_2/p)$	1980	Kenneth S. Williams
+	On the least quadratic non-residue of a prime p ≡ 3 (mod 4).	1980	Richard H. Hudson Kenneth S. Williams
+	The quartic characters of certain quadratic units	1980	Philip Leonard Kenneth S. Williams
+	An Artin Character and Represenattions of Primes by Binary Quadratic Forms.	1980	Pierre Kaplan Kenneth S. Williams
+ PDF Chat	The quadratic and quartic character of certain quadratic units. II	1979	Philip Leonard Kenneth S. Williams
+ PDF Chat	An Octic Reciprocity Law of Scholz Type	1979	Duncan A. Buell Kenneth S. Williams
+ PDF Chat	The class number of<i>Q</i>(√−<i>p</i>) modulo 4, for<i>p</i>≡ 3(mod4) a prime	1979	Kenneth S. Williams
+ PDF Chat	Evaluation of Character Sums Connected with Elliptic Curves	1979	Kenneth S. Williams
+ PDF Chat	Note on Brewer’s character sum	1978	Kenneth S. Williams
+	Is There an Octic Reciprocity Law of Scholz Type?	1978	Duncan A. Buell Kenneth S. Williams
+	Is there an Octic Reciprocity Law of Scholz Type?	1978	Duncan A. Buell Kenneth S. Williams
+ PDF Chat	Maximal residue difference sets modulo 𝑝	1978	Duncan A. Buell Kenneth S. Williams
+	On Scholz's Reciprocity Law	1977	Kenneth S. Williams
+ PDF Chat	Note on Burde's Rational Biquadratic Reciprocity Law	1977	Kenneth S. Williams
+ PDF Chat	On Scholz’s reciprocity law	1977	Kenneth S. Williams
+	Explicit forms of Kummer's complementary theorems to his law of quintic reciprocity.	1976	Kenneth S. Williams
+ PDF Chat	Explicit Criteria for Quintic Residuacity	1976	Kenneth S. Williams
+	Note on a result of Barrucand and Cohn.	1976	Kenneth S. Williams
+ PDF Chat	On the Supplement to the Law of Biquadratic Reciprocity	1976	Kenneth S. Williams
+	Note on a Result of Kaplan	1976	Kenneth S. Williams
+	The Quadratic Character of 2 mod <i>p</i>	1976	Kenneth S. Williams
+	The Quadratic Character of 2 mod p	1976	Kenneth S. Williams
+ PDF Chat	Note on a result of Kaplan	1976	Kenneth S. Williams
+ PDF Chat	On the supplement to the law of biquadratic reciprocity	1976	Kenneth S. Williams
+ PDF Chat	On Euler’s criterion for quintic nonresidues	1975	Kenneth S. Williams
+	2 AS A NINTH POWER (MOD p)	1975	Kenneth S. Williams
+	Note on cubics over GF(2n) and GF(3n)	1975	Kenneth S. Williams
+ PDF Chat	The Cyclotomic Numbers of Order Seven	1975	Philip Leonard Kenneth S. Williams
+	Jacobi sums and a theorem of Brewer	1975	Philip Leonard Kenneth S. Williams
+	Note on a cubic character sum	1975	Kenneth S. Williams
+ PDF Chat	On Euler’s criterion for cubic nonresidues	1975	Kenneth S. Williams
+	On Euler's Criterion for Cubic Nonresidues	1975	Kenneth S. Williams
+	Note on Non-Euclidean Principal Ideal Domains	1975	Kenneth S. Williams
+	Note on Non-Euclidean Principal Ideal Domains	1975	Kenneth S. Williams
+ PDF Chat	Note on Integers Representable by Binary Quadratic Forms	1975	Kenneth S. Williams
+ PDF Chat	Note on the Supplement to the Law of Cubic Reciprocity	1975	Kenneth S. Williams
+	Table of Solutions (x, u, v, w) of the Diophantine System 16p = x 2 + 50u 2 + 50v 2 + 125w 2 , xw = v 2 - 4uv - u 2 , x ≡1 (mod 5) for Primes p < 10000, p ≡1 (mod 5)	1975	Emma Lehmer Kenneth S. Williams B. Löwe
+ PDF Chat	Note on the supplement to the law of cubic reciprocity	1975	Kenneth S. Williams
+	Note on Cubits over GF(2') and GF(3")*	1975	Kenneth S. Williams
+	Handbook of estimates in the theory of numbers	1975	Blair K. Spearman Kenneth S. Williams
+ PDF Chat	Elementary treatment of a quadratic partition of primes $p \equiv 1 (\mathrm{mod} 7)$	1974	Kenneth S. Williams
+	On certain sums of fractional parts	1974	J. M. Gandhi Kenneth S. Williams
+	A Quadratic Partition of Primes ≡1 (mod 7)	1974	Kenneth S. Williams
+	Merten's theorem for arithmetic progressions	1974	Kenneth S. Williams
+ PDF Chat	Note on a Paper of S. Uchiyama	1974	Brian Mortimer Kenneth S. Williams
+ PDF Chat	A quadratic partition of primes ≡1 (𝑚𝑜𝑑7)	1974	Kenneth S. Williams
+	Representability of binary quadratic forms over a Bézout domain	1973	Philip Leonard Kenneth S. Williams
+ PDF Chat	The Kloosterman Sum Revisited	1973	Kenneth S. Williams
+	Another Proof of a Theorem of Niven	1973	Kenneth S. Williams
+	Quartics over GF(2 n )	1972	Philip Leonard Kenneth S. Williams
+	On the number of distinguished representations of a group element	1972	D. Jacobson Kenneth S. Williams
+ PDF Chat	Representation of a Binary Quadratic Form as a Sum of Two Squares	1972	Kenneth S. Williams
+ PDF Chat	Quartics over ${\rm GF}(2\sp{n})$	1972	Philip Leonard Kenneth S. Williams
+ PDF Chat	Representation of a binary quadratic form as a sum of two squares	1972	Kenneth S. Williams
+	Note on Dickson’s permutation polynomials	1971	Kenneth S. Williams
+ PDF Chat	Forms Representable by an Integral Positive-Definite Binary Quadratic Form.	1971	Kenneth S. Williams
+	On ∑<sup>∞</sup><sub>n = 1</sub> (1/n<sup>2k</sup>)	1971	Kenneth S. Williams
+	Small solutions of the congruence	1971	Kenneth S. Williams
+ PDF Chat	On the solution of linear G<i>.</i>C<i>.</i>D<i>.</i>equations	1971	David Jacobson Kenneth S. Williams
+ PDF Chat	Note on the Number of Solutions of <i>f(x<sub>1</sub>) = f(x<sub>2</sub>) = … =f(x<sub>r</sub>)</i> over a Finite Field	1971	Kenneth S. Williams
+ PDF Chat	Note on the Kloosterman Sum	1971	Kenneth S. Williams
+	Shorter Notes: Note on a Theorem of Pall	1971	Kenneth S. Williams
+	A distribution property of the solutions of a congruence modulo a large prime	1971	Kenneth S. Williams
+ PDF Chat	Note on the Kloosterman sum	1971	Kenneth S. Williams
+ PDF Chat	Note on Salié’s sum	1971	Kenneth S. Williams
+	A Class of Character Sums	1971	Kenneth S. Williams
+	Note on ∫<sup>∞</sup><sub>0</sub> (sin x/x)dx	1971	Kenneth S. Williams
+ PDF Chat	Integers of Biquadratic Fields	1970	Kenneth S. Williams
+	On a Result of Libri and Lebesgue	1970	Kenneth S. Williams
+ PDF Chat	On Two Conjectures of Chowla	1969	Kenneth S. Williams
+ PDF Chat	Note on Factorable Polynomials	1969	Kenneth S. Williams
+	The distribution of solutions of congruences: <i>corrigendum and addendum</i>	1969	J. H. H. Chalk Kenneth S. Williams
+ PDF Chat	Small Solutions of the Congruence ax<sup>2</sup> + by<sup>2</sup> ≡ c(mod k)	1969	Kenneth S. Williams
+ PDF Chat	Polynomials with Irreducible Factors of Specified Degree	1969	Kenneth S. Williams
+ PDF Chat	Collected Papers of G. H. Hardy, Volume 2. Edited by a committee appointed by the London Mathematical Society, Oxford 1967, 702 pages. Can $18.	1969	Kenneth S. Williams
+ PDF Chat	Distinct Values of a Polynomial in Subsets of a Finite Field	1969	Kenneth S. Williams
+	Quadratic Polynomials with the Same Residues	1968	Kenneth S. Williams
+	Cubic polynomials with the same residues (mod <i>p</i>)	1968	K. McCann Kenneth S. Williams
+ PDF Chat	On Exceptional Polynomials	1968	Kenneth S. Williams
+ PDF Chat	On a Theorem of Niven	1968	Kenneth S. Williams
+ PDF Chat	Note on Pairs of Consecutive Residues of Polynomials	1968	Kenneth S. Williams
+ PDF Chat	On Extremal Polynomials	1967	Kenneth S. Williams
+ PDF Chat	On General Polynomials	1967	Kenneth S. Williams
+ PDF Chat	On A Theorem of Niven	1967	Kenneth S. Williams
+	A Sum of Fractional Parts	1967	Kenneth S. Williams
+ PDF Chat	The distribution of the residues of a quartic polynomial	1967	K. McCann Kenneth S. Williams
+ PDF Chat	On the Residues of a Cubic Polynomial (Mod p )	1967	K. McCann Kenneth S. Williams
+ PDF Chat	Pairs of Consecutive Residues of Polynomials	1967	Kenneth S. Williams
+ PDF Chat	Eisenstein's Criteria for Absolute Irreducibility Over a Finite Field	1966	Kenneth S. Williams
+	On the least non-residue of a quartic polynomial	1966	Kenneth S. Williams
+	On the Number of Solutions of a Congurence	1966	Kenneth S. Williams
+	The distribution of solutions of congruences	1965	J. H. H. Chalk Kenneth S. Williams
+	A Note on the Quadratic Form <i>ax</i><sup>2</sup> + 2<i>bxy</i> + <i>cy</i><sup>2</sup>	1964	Kenneth S. Williams
+	3094. On the factorization of the first four composite Merdsenne numbers	1964	Kenneth S. Williams
+	3094. On the Factorization of the First Four Composite Mersenne Numbers	1964	Kenneth S. Williams
+	3039. On the divisibility of F<sub>6</sub> by 274,177	1962	Kenneth S. Williams
+	3029. A generalisation of Cardan's solution of the cubic	1962	Kenneth S. Williams

Common Coauthors

Coauthor	Papers Together
Blair K. Spearman	63
Şaban Alaca	35
Ayşe Alaca	29
Pierre Kaplan	18
Kenneth Hardy	17
Richard H. Hudson	17
Philip Leonard	16
James G. Huard	13
Lerna Pehlivan	10
Jerzy Urbanowicz	7
Duncan A. Buell	6
Christian Friesen	5
Zafer Selçuk Aygin	5
Joseph B. Muskat	5
Mathieu F. Lemire	5
David R. Richman	4
Florian Münkel	4
Greg Doyle	4
Faruk Uygul	4
Robert H. E. Hudson	4
Habib Muzaffar	3
Melisa J. Lavallee	3
Daniël Eloff	3
Qiduan Yang	3
Bruce C. Berndt	3
Vlastimil Dlab	3
K. J. Hardy	3
K. McCann	3
Zhiming M. Ou	2
Nicholas Buck	2
Zhang Nan-Yue	2
Nicholas Buck	2
Eric Driver	2
N. M. Holtz	2
Nan Yue Zhang	2
Alan K. Silvester	2
Zhang Yue	2
Zhi-Hong Sun	2
Edward Dobrowolski	2
J. H. H. Chalk	2
Ronald J. Evans	2
Lones Smith	2
Byungchan Kim	1
Emma Lehmer	1
Blair K. Spear	1
D. LIU	1
Noriaki Kimura	1
Richard Hudson	1
Noriaki Kimura	1
Glenn J. Fox	1

Commonly Cited References

Action	Title	Year	Authors	# of times referenced
+	Number theory in the spirit of Ramanujan	2007	Bruce C. Berndt	17
+	Number Theory in the Spirit of Ramanujan	2006	Bruce C. Berndt	16
+ PDF Chat	Elementary and Analytic Theory of Algebraic Numbers	1991	Peter Shiu Władysław Narkiewicz	15
+	Ramanujan’s Notebooks: Part V	1997	Bruce C. Berndt	15
+ PDF Chat	Nineteen quaternary quadratic forms	2007	Ayşe Alaca Şaban Alaca Mathieu F. Lemire Kenneth S. Williams	14
+ PDF Chat	The Chowla-Selberg formula for genera	1995	James G. Huard Pierre Kaplan Kenneth S. Williams	13
+ PDF Chat	On the two-dimensional theta functions of the Borweins	2006	Ayşe Alaca Şaban Alaca Kenneth S. Williams	12
+	Binary Quadratic Forms	1989	Duncan A. Buell	11
+	A Course in Computational Algebraic Number Theory	1993	Henri Cohen	9
+	An Introduction to the Theory of Numbers. By G. H. Hardy and E. M. Wright. 2nd edition. Pp. xvi, 407 25s. 1945. (Oxford)	1946	T. A. A. B.	9
+	Criteria for cubic and quartic residuacity	1958	Emma Lehmer	9
+	Cyclotomy, Higher Congruences, and Waring's Problem	1935	L. E. Dickson	9
+ PDF Chat	The convolution sum ∑<sub>m<n⁄8</sub>σ(m) σ(n−8m)	2006	Kenneth S. Williams	9
+ PDF Chat	Solving solvable quintics	1991	David S. Dummit	9
+	An Introduction to the Theory of Numbers.	1961	W. J. LeVeque Ivan Niven Herbert S. Zuckerman	9
+ PDF Chat	An introduction to the theory of numbers	1960	G. H. Hardy	9
+	The power of 2 dividing the class-number of a binary quadratic discriminant	1973	Ezra Brown	9
+	Fundamenta nova theoriae functionum ellipticarum	2013	Carl Gustav Jacob Jacobi	8
+	THETA FUNCTION IDENTITIES AND REPRESENTATIONS BY CERTAIN QUATERNARY QUADRATIC FORMS	2008	Ayşe Alaca Şaban Alaca Mathieu F. Lemire Kenneth S. Williams	8
+	Integral Bases for Quartic Fields with Quadratic Subfields	1995	James G. Huard Blair K. Spearman Kenneth S. Williams	8
+	Introduction to number theory	2009	Luo Hua	8
+	History of the Theory of Numbers	1919	L. E. Dickson	8
+ PDF Chat	CONVOLUTION SUMS INVOLVING THE DIVISOR FUNCTION	2004	Nathalie Cheng Kenneth S. Williams	7
+ PDF Chat	On Euler's criterion	1959	Emma Lehmer	7
+	Ramanujan's Notebooks	1985	Bruce C. Berndt	7
+	Spinor genera of binary quadratic forms	1973	Dennis R. Estes Gordon Pall	7
+	Characterization of Solvable Quintics <i>x</i><sup>5</sup> + <i>ax</i> + <i>b</i>	1994	Blair K. Spearman Kenneth S. Williams	7
+	Classical theorems on quadratic residues	1976	Bruce C. Berndt	7
+	THE CONVOLUTION SUM $\sum\limits_{m<n/9}\sigma(m)\sigma(n-9m)$	2005	Kenneth S. Williams	6
+	Bounds for exponential sums	1957	L. Carlitz Saburoô Uchiyama	6
+	On a conjecture of hasse concerning multiplicative relations of Gaussian sums	1966	Kōichi Yamamoto	6
+	Introduction to Number Theory	1982	Hua Loo Keng	6
+ PDF Chat	Calculation of the Class Numbers of Imaginary Cyclic Quartic Fields	1987	Kenneth Hardy Robert H. E. Hudson David R. Richman Kenneth S. Williams N. M. Holtz	6
+ PDF Chat	The cyclotomic numbers of order twelve	1960	A. Whiteman	6
+ PDF Chat	Multiplicities of dihedral discriminants	1992	Daniel C. Mayer	6
+	On Epstein's Zeta-function.	1967	S. Chowla Atle Selberg	6
+	On the quadratic character of some quadratic surds.	1971	Emma Lehmer	6
+	THE NUMBER OF REPRESENTATIONS OF A POSITIVE INTEGER BY CERTAIN QUATERNARY QUADRATIC FORMS	2009	Ayşe Alaca Şaban Alaca Mathieu F. Lemire Kenneth S. Williams	6
+	Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen.	1935	H. Davenport Helmut Hasse	6
+ PDF Chat	Effective determination of the decomposition of the rational primes in a cubic field	1983	Pascual Llorente Enric Nart	6
+	Note on primes of type x2 + 32y2, class number, and residuacity.	1969	Harvey Cohn Pierre Barrucand	6
+	Advanced analytic number theory	1980	Carl Ludwig Siegel	5
+ PDF Chat	On the evaluation of Brewer’s character sums	1972	Reinaldo E. Giudici Joseph B. Muskat Stanley F. Robinson	5
+ PDF Chat	Ramanujan’s theories of elliptic functions to alternative bases	1995	Bruce C. Berndt Saurabh Bhargava Frank G. Garvan	5
+	15 Introduction to the theory of functions of a complex variable	2022	Vemuri Balakotaiah Ram R. Ratnakar	5
+ PDF Chat	The cyclotomic numbers of order fourteen	1966	Joseph B. Muskat	5
+ PDF Chat	The quadratic and quartic character of certain quadratic units. I	1977	Philip Leonard Kenneth Williams	5
+	Cyclotomy, Higher Congruences, and Waring's Problem II	1935	L. E. Dickson	5
+ PDF Chat	Seven octonary quadratic forms	2008	Ayşe Alaca Şaban Alaca Kenneth S. Williams	5
+ PDF Chat	The cyclotomic numbers of order twenty	1970	Joseph B. Muskat A. Whiteman	5