Daniel Shanks

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All published works

Action	Title	Year	Authors
+	Calculation of π to 100,000 Decimals	2004	Daniel Shanks John W. Wrench
+	On a Sequence Arising in Series for π	2004	Morris Newman Daniel Shanks
+	On a Sequence Arising in Series for π	2000	Morris Newman Daniel Shanks
+	Calculation of π to 100,000 Decimals	2000	Daniel Shanks John W. Wrench
+ PDF Chat	Zeros of 2-adic 𝐿-functions and congruences for class numbers and fundamental units	1999	Daniel Shanks Patrick J. Sime Lawrence C. Washington
+	Calculation of π to 100,000 Decimals	1997	Daniel Shanks John W. Wrench
+	48 More Solutions of Martin Davis's Quaternary Quartic Equation	1995	Daniel Shanks Samuel S. Wagstaff
+ PDF Chat	48 more solutions of Martin Davis’s quaternary quartic equation	1995	Daniel Shanks Samuel S. Wagstaff
+	Solved and Unsolved Problems in Number Theory.	1987	H. C. W. Daniel Shanks
+	Fast Primality Tests for Numbers Less Than 50 ⋅10 9	1986	G. C. Kurtz Daniel Shanks Hywel C Williams
+	On a sequence arising in series for pi (with an appendix by D. Zagier).	1984	Morris Newman Daniel Shanks
+ PDF Chat	On a sequence arising in series for 𝜋	1984	Morris Newman Daniel Shanks
+	On a Sequence Arising in Series for π	1984	Morris Newman Daniel Shanks
+	On a Sequence Arising in Series for &#960	1984	Morris Newman Daniel Shanks
+ PDF Chat	Strong Primality Tests that are Not Sufficient	1982	William H. Adams Daniel Shanks
+	Dihedral quartic approximations and series for π	1982	Daniel Shanks
+ PDF Chat	Strong primality tests that are not sufficient	1982	William W. Adams Daniel Shanks
+	Solved and Unsolved Problems in Number Theory.	1982	Paul T. Bateman Daniel Shanks
+ PDF Chat	Gunderson’s function in Fermat’s last theorem	1981	Daniel Shanks Hywel C Williams
+	Gunderson's Function in Fermat's Last Theorem	1981	Daniel Shanks H. C. Williams
+ PDF Chat	Simple groups of square order and interesting sequence of primes	1980	Morris Newman Daniel Shanks H. H. Williams
+	A Note on Class-Number One in Pure Cubic Field	1979	Hywel C Williams Daniel Shanks
+	Quadratic Fields With 3-Rank Equal to 4	1979	Fredéric Diaz Daniel Shanks H. C. Williams
+ PDF Chat	Quadratic fields with 3-rank equal to 4	1979	Fredéric Diaz Daniel Shanks Hywel C Williams
+ PDF Chat	A note on class-number one in pure cubic fields	1979	Hywel C Williams Daniel Shanks
+ PDF Chat	Corrigenda: “Gauss’s ternary form reduction and the 2-Sylow subgroup” (Math. Comp. 25 (1971), 837–853)	1978	Daniel Shanks
+	Three Tables Concerning the Parity of the Partition Numbers p(n) for n < 2040000	1976	D. S. T. R. Parkin Daniel Shanks
+ PDF Chat	Table errata: “Regular continued fractions for 𝜋 and 𝛾”, (Math. Comp. 25 (1971), 403); “Rational approximations to 𝜋” (ibid. 25 (1971), 387–392) by K. Y. Choong, D. E. Daykin and C. R. Rathbone	1976	Daniel Shanks
+ PDF Chat	Corrigendum: “Class groups of the quadratic fields found by F. Diaz y Diaz” (Math. Comp. 30 (1976), no. 133, 173–178)	1976	Daniel Shanks
+ PDF Chat	Class groups of the quadratic fields found by F. Diaz y Diaz	1976	Daniel Shanks
+ PDF Chat	Corrigendum: “Calculation and applications of Epstein zeta functions” (Math. Comp. 29 (1975), 271–287)	1976	Daniel Shanks
+ PDF Chat	Class Groups of the Quadratic Fields Found by F. Diaz y Diaz	1976	Daniel Shanks
+	Non-Hypotenuse Numbers	1975	Daniel Shanks
+ PDF Chat	Calculation and applications of Epstein zeta functions	1975	Daniel Shanks
+ PDF Chat	Corrigenda: “Calculation and applications of Epstein zeta functions” (Math. Comp. 29 (1975), 271–287)	1975	Daniel Shanks
+	Sums of Reciprocals to 1,000,000	1975	D. S. J. W. W. Daniel Shanks John W. Wrench
+ PDF Chat	Calculation and Applications of Epstein Zeta Functions	1975	Daniel Shanks
+ PDF Chat	The Simplest Cubic Fields	1974	Daniel Shanks
+ PDF Chat	On the 3-rank of quadratic fields and the Euler product	1974	Carol Neild Daniel Shanks
+ PDF Chat	Brun’s constant	1974	Daniel Shanks John W. Wrench
+ PDF Chat	The simplest cubic fields	1974	Daniel Shanks
+	On the 3-Rank of Quadratic Fields and the Euler Product	1974	Carol Neild Daniel Shanks
+	An Observation on Fibonacci Primitive Roots	1973	Daniel Shanks Larry Taylor
+ PDF Chat	Corrigenda: “Quadratic fields with four invariants divisible by 3” (Math. Comp. 27 (1973), 183–187) by Daniel Shanks and Richard Serafin	1973	Daniel Shanks
+ PDF Chat	Quadratic fields with four invariants divisible by 3	1973	Daniel Shanks Richard Serafin
+ PDF Chat	Table errata: “A note on the Catalan-Dickson conjecture” (Math. Comp. 27 (1973), 189–192)	1973	Daniel Shanks
+	Quadratic Fields with Four Invariants Divisible by 3	1973	Daniel Shanks Richard Serafin
+	Systematic examination of Littlewood’s bounds on 𝐿(1,𝜒)	1973	Daniel Shanks
+	New types of quadratic fields having three invariants divisible by 3	1972	Daniel Shanks
+	A necessary and sufficient condition for primality, and its source	1972	Henry B. Mann Daniel Shanks
+	Fibonacci Primitive Roots	1972	Daniel Shanks
+ PDF Chat	Bateman’s constants reconsidered and the distribution of cubic residues	1972	Daniel Shanks Mohan Lal
+	Bateman's Constants Reconsidered and the Distribution of Cubic Residues	1972	Daniel Shanks Mohan Lal
+	Gauss's Ternary Form Reduction and the 2-Sylow Subgroup	1971	Daniel Shanks
+ PDF Chat	Corrigendum to: “On generating infinitely many solutions of the Diophantine equation 𝐴⁶+𝐵⁶+𝐶⁶=𝐷⁶+𝐸⁶+𝐹⁶” (Math. Comp. 24 (1970), 453–454) by Simcha Brudno	1971	Daniel Shanks
+ PDF Chat	Corrigendum to: “Statistical study of digits of some square roots of integers in various bases” (Math. Comp. 24 (1970), 455–473) by W. A. Beyer, N. Metropolis and J. R. Neergaard	1971	Daniel Shanks
+ PDF Chat	Integer sequences having prescribed quadratic character	1970	D. H. Lehmer Emma Lehmer Daniel Shanks
+	Calculation of e to 100,000 Decimals	1969	Daniel Shanks John W. Wrench
+ PDF Chat	Corrigenda: “Solutions of the Diophantine equations 𝑥²+𝑦²=𝑙², 𝑦²+𝑧²=𝑚², 𝑧²+𝑥²=𝑛²” (Math. Comp. 20 (1966), 144–147) by M. Lal and W. J. Blundon	1969	Daniel Shanks
+ PDF Chat	On Gauss’s class number problems	1969	Daniel Shanks
+	On Gauss's Class Number Problems	1969	Daniel Shanks
+ PDF Chat	Corrigenda to: “Generalized Euler and class numbers “	1968	Daniel Shanks
+ PDF Chat	Generalized Euler and Class Numbers	1967	Daniel Shanks
+	Lal's Constant and Generalizations	1967	Daniel Shanks
+	On the Distribution of Parity in the Partition Function	1967	T. R. Parkin Daniel Shanks
+ PDF Chat	On the distribution of Mersenne divisors	1967	Daniel Shanks Sidney Kravitz
+ PDF Chat	On the distribution of parity in the partition function	1967	T. R. Parkin Daniel Shanks
+ PDF Chat	Lal’s constant and generalizations	1967	Daniel Shanks
+ PDF Chat	Generalized Euler and class numbers	1967	Daniel Shanks
+ PDF Chat	On the Distribution of Mersenne Divisors	1967	Daniel Shanks Sidney Kravitz
+	Variations on a Theorem of Landau. Part I	1966	Daniel Shanks Larry P. Schmid
+	Questions Concerning Khintchine's Constant and the Efficient Computation of Regular Continued Fractions	1966	John W. Wrench Daniel Shanks
+ PDF Chat	Variations on a theorem of Landau. I	1966	Daniel Shanks Larry P. Schmid
+ PDF Chat	Questions concerning Khintchine’s constant and the efficient computation of regular continued fractions	1966	John W. Wrench Daniel Shanks
+ PDF Chat	On Maximal Gaps between Successive Primes	1964	Daniel Shanks
+ PDF Chat	An analytic criterion for the existence of infinitely many primes of the form $\frac{1}{2}(n^{2}+1)$	1964	Daniel Shanks
+	Solved and Unsolved Problems in Number Theory.	1964	W. J. LeVeque Daniel Shanks
+ PDF Chat	Polylogarithms, Dirichlet Series, and Certain Constants	1964	Daniel Shanks
+ PDF Chat	On maximal gaps between successive primes	1964	Daniel Shanks
+ PDF Chat	Polylogarithms, Dirichlet series, and certain constants	1964	Daniel Shanks
+ PDF Chat	The second-order term in the asymptotic expansion of 𝐵(𝑥)	1964	Daniel Shanks
+	Solved and Unsolved Problems in Number Theory	1963	Richard Bellman Daniel Shanks
+ PDF Chat	The Calculation of Certain Dirichlet Series	1963	Daniel Shanks John W. Wrench
+ PDF Chat	Supplementary Data and Remarks Concerning a Hardy-Littlewood Conjecture	1963	Daniel Shanks
+ PDF Chat	The calculation of certain Dirichlet series	1963	Daniel Shanks John W. Wrench
+ PDF Chat	Supplementary data and remarks concerning a Hardy-Littlewood conjecture	1963	Daniel Shanks
+	Calculation of π to 100,000 Decimals	1962	Daniel Shanks John W. Wrench
+	On Numbers of the Form n 4 + 1	1961	Daniel Shanks
+ PDF Chat	On numbers of the form 𝑛⁴+1	1961	Daniel Shanks
+	On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n 2 + a	1960	Daniel Shanks
+ PDF Chat	A Note on Gaussian Twin Primes	1960	Daniel Shanks
+ PDF Chat	A note on Gaussian twin primes	1960	Daniel Shanks
+ PDF Chat	On the conjecture of Hardy & Littlewood concerning the number of primes of the form 𝑛²+𝑎	1960	Daniel Shanks
+	Quadratic Residues and the Distribution of Primes	1959	Daniel Shanks
+	A Sieve Method for Factoring Numbers of the Form n 2 + 1	1959	Daniel Shanks
+	A sieve method for factoring numbers of the form 𝑛²+1	1959	Daniel Shanks
+ PDF Chat	Quadratic residues and the distribution of primes	1959	Daniel Shanks
+ PDF Chat	Two theorems of Gauss	1958	Daniel Shanks
+	THE NUMERICAL SOLUTION OF THE HEAT CONDUCTION EQUATION OCCURRING IN THE THEORY OF THERMAL EXPLOSIONS	1956	Julius W. Enig Daniel Shanks Raymond W. Southworth
+	Non‐linear Transformations of Divergent and Slowly Convergent Sequences	1955	Daniel Shanks
+ PDF Chat	A Short Proof of an Identity of Euler	1951	Daniel Shanks
+ PDF Chat	A short proof of an identity of Euler	1951	Daniel Shanks

Common Coauthors

Coauthor	Papers Together
John W. Wrench	11
Morris Newman	7
Hywel C Williams	5
T. R. Parkin	3
Richard Serafin	2
H. C. Williams	2
Fredéric Diaz	2
Sidney Kravitz	2
Carol Neild	2
D. S.	2
Larry P. Schmid	2
Samuel S. Wagstaff	2
Mohan Lal	2
G. C. Kurtz	1
Julius W. Enig	1
H. C. W.	1
Emma Lehmer	1
Lawrence C. Washington	1
Henry B. Mann	1
Richard Bellman	1
Patrick J. Sime	1
D. H. Lehmer	1
Paul T. Bateman	1
H. H. Williams	1
Raymond W. Southworth	1
William H. Adams	1
J. W. W.	1
W. J. LeVeque	1
Larry Taylor	1
William W. Adams	1

Commonly Cited References

Action	Title	Year	Authors	# of times referenced
+ PDF Chat	On the conjecture of Hardy & Littlewood concerning the number of primes of the form 𝑛²+𝑎	1960	Daniel Shanks	13
+ PDF Chat	The Calculation of Certain Dirichlet Series	1963	Daniel Shanks John W. Wrench	10
+ PDF Chat	The calculation of certain Dirichlet series	1963	Daniel Shanks John W. Wrench	10
+	Variations on a Theorem of Landau. Part I	1966	Daniel Shanks Larry P. Schmid	7
+ PDF Chat	Supplementary Data and Remarks Concerning a Hardy-Littlewood Conjecture	1963	Daniel Shanks	6
+	Quadratic Fields with Four Invariants Divisible by 3	1973	Daniel Shanks Richard Serafin	6
+ PDF Chat	Variations on a theorem of Landau. I	1966	Daniel Shanks Larry P. Schmid	6
+	A Sieve Method for Factoring Numbers of the Form n 2 + 1	1959	Daniel Shanks	5
+ PDF Chat	A heuristic asymptotic formula concerning the distribution of prime numbers	1962	Paul T. Bateman Roger A. Horn	5
+	Über die Beziehung der Klassenzahlen quadratischer Körper zueinander.	1932	Arnold Scholz	5
+	New types of quadratic fields having three invariants divisible by 3	1972	Daniel Shanks	5
+	On the 3-Rank of Quadratic Fields and the Euler Product	1974	Carol Neild Daniel Shanks	5
+ PDF Chat	A Heuristic Asymptotic Formula Concerning the Distribution of Prime Numbers	1962	Paul T. Bateman Roger A. Horn	5
+	Handbuch der Lehre von der Verteilung der Primzahlen	1911	Edmund Landau	5
+ PDF Chat	Quadratic fields with four invariants divisible by 3	1973	Daniel Shanks Richard Serafin	5
+ PDF Chat	Calculation and Applications of Epstein Zeta Functions	1975	Daniel Shanks	5
+	Solved and Unsolved Problems in Number Theory	1963	Richard Bellman Daniel Shanks	5
+ PDF Chat	Quadratic residues and the distribution of primes	1959	Daniel Shanks	5
+ PDF Chat	Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes	1923	G. H. Hardy J. E. Littlewood	4
+ PDF Chat	A note on Gaussian twin primes	1960	Daniel Shanks	4
+	Evaluation of Artin's Constant and the Twin-Prime Constant	1961	John W. Wrench	4
+	Quadratic Residues and the Distribution of Primes	1959	Daniel Shanks	4
+ PDF Chat	Supplementary data and remarks concerning a Hardy-Littlewood conjecture	1963	Daniel Shanks	4
+ PDF Chat	Generalized Euler and Class Numbers	1967	Daniel Shanks	4
+ PDF Chat	The distribution of integers represented by binary quadratic forms	1943	Gordon Pall	4
+ PDF Chat	waring's problem for algebraic number fields and primes of the form $(p^{r} - 1)/(p^{r} - 1)$	1962	Paul T. Bateman Rosemarie M. Stemmler	3
+ PDF Chat	Strong primality tests that are not sufficient	1982	William W. Adams Daniel Shanks	3
+ PDF Chat	The second-order term in the asymptotic expansion of 𝐵(𝑥)	1964	Daniel Shanks	3
+ PDF Chat	Calculation and applications of Epstein zeta functions	1975	Daniel Shanks	3
+ PDF Chat	Evaluation of Artin’s constant and the twin-prime constant	1961	John W. Wrench	3
+	A sieve method for factoring numbers of the form 𝑛²+1	1959	Daniel Shanks	3
+	Dihedral quartic approximations and series for π	1982	Daniel Shanks	3
+	History of the Theory of Numbers	1919	L. E. Dickson	3
+	Modular functions in analytic number theory	1970	Marvin I. Knopp	3
+ PDF Chat	A Short Proof of an Identity of Euler	1951	Daniel Shanks	3
+ PDF Chat	Polylogarithms, Dirichlet Series, and Certain Constants	1964	Daniel Shanks	3
+ PDF Chat	On the 3-rank of quadratic fields and the Euler product	1974	Carol Neild Daniel Shanks	3
+ PDF Chat	Polylogarithms, Dirichlet series, and certain constants	1964	Daniel Shanks	3
+ PDF Chat	On numbers of the form 𝑛⁴+1	1961	Daniel Shanks	3
+	Note on Partitions Modulo 5	1967	Morris Newman	3
+	Tables Concerning the Distribution of Primes up to 37 Millions	1959	J. L. Selfridge D. H. Lehmer	3
+ PDF Chat	Cyclotomy with short periods	1983	D. H. Lehmer Emma Lehmer	3
+	The Number of Subgroups of the Classical Modular Group of Index N.	1978	D. S. A. O. L. Atkin	3
+	On the Exponent of the Ideal Class Groups of Complex Quadratic Fields	1972	David W. Boyd H. Kisilevsky	3
+ PDF Chat	Real zeros of the Dedekind zeta function of an imaginary quadratic field	1968	Marc Edmon Low	2
+ PDF Chat	An introduction to the theory of numbers	1960	G. H. Hardy	2
+	Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante	1912	Paul Bernays	2
+ PDF Chat	On the exponent of the ideal class groups of complex quadratic fields.	1972	David W. Boyd Hershy Kisilevsky	2
+	Remarks on principal factors in a relative cubic field	1971	Pierre Barrucand Harvey Cohn	2
+ PDF Chat	Generalized Euler and class numbers	1967	Daniel Shanks	2