The mathematical work of Morgan Ward

Lehmer, D. H.

doi:10.1090/S0025-5718-1993-1182245-3

The mathematical work of Morgan Ward
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Math. Comp. 61 (1993), 307-311 Request permission

References

Morgan Ward, A generalization of recurrents, Bull. Amer. Math. Soc. 33 (1927), no. 4, 477–492. MR 1561403, DOI 10.1090/S0002-9904-1927-04410-X

General arithmetic

13

Postulates for an abstract arithmetic

14

Morgan Ward, A Simplification of Certain Problems in Arithmetical Division, Amer. Math. Monthly 35 (1928), no. 1, 9–14. MR 1521339, DOI 10.2307/2298578
Morgan Ward, On Certain Functional Relations, Amer. Math. Monthly 36 (1929), no. 8, 426–431. MR 1521822, DOI 10.2307/2299940

Certain expansions involving doubly infinite series

30

Morgan Ward, A certain class of polynomials, Ann. of Math. (2) 31 (1930), no. 1, 43–51. MR 1502917, DOI 10.2307/1968138

A generalization of a familiar theorem concerning prime numbers

5

The reversion of a power series

54

Morgan Ward, Postulates for the inverse operations in a group, Trans. Amer. Math. Soc. 32 (1930), no. 3, 520–526. MR 1501550, DOI 10.1090/S0002-9947-1930-1501550-7
Morgan Ward, The characteristic number of a sequence of integers satisfying a linear recursion relation, Trans. Amer. Math. Soc. 33 (1931), no. 1, 153–165. MR 1501582, DOI 10.1090/S0002-9947-1931-1501582-X
Morgan Ward, The algebra of recurring series, Ann. of Math. (2) 32 (1931), no. 1, 1–9. MR 1502975, DOI 10.2307/1968408
Morgan Ward, Some arithmetical properties of sequences satisfying a linear recursion relation, Ann. of Math. (2) 32 (1931), no. 4, 734–738. MR 1503026, DOI 10.2307/1968316
Morgan Ward, The distribution of residues in a sequence satisfying a linear recursion relation, Trans. Amer. Math. Soc. 33 (1931), no. 1, 166–190. MR 1501583, DOI 10.1090/S0002-9947-1931-1501583-1
Morgan Ward, Conditions for the solubility of the Diophantine equation $x^2-My^2=-1$, Trans. Amer. Math. Soc. 33 (1931), no. 3, 712–718. MR 1501611, DOI 10.1090/S0002-9947-1931-1501611-3
Morgan Ward, The linear form of numbers represented by a homogeneous polynomial in any number of variables, Ann. of Math. (2) 33 (1932), no. 2, 324–326. MR 1503056, DOI 10.2307/1968334

On the behaviour of non-static models of the universe when the cosmological term is omitted

39

Morgan Ward, A Type of Multiplicative Diophantine System, Amer. J. Math. 55 (1933), no. 1-4, 67–76. MR 1506944, DOI 10.2307/2371110

A property of recurring series

19

Morgan Ward, The cancellation law in the theory of congruences to a double modulus, Trans. Amer. Math. Soc. 35 (1933), no. 1, 254–260. MR 1501682, DOI 10.1090/S0002-9947-1933-1501682-6
Morgan Ward, The arithmetical theory of linear recurring series, Trans. Amer. Math. Soc. 35 (1933), no. 3, 600–628. MR 1501705, DOI 10.1090/S0002-9947-1933-1501705-4
Morgan Ward, Questions, Discussions, and Notes: A Certain Class of Trigonometric Integrals, Amer. Math. Monthly 40 (1933), no. 6, 340–346. MR 1522833, DOI 10.2307/2301523
Morgan Ward, The Representation of Stirling’s Numbers and Stirling’s Polynomials as Sums of Factorials, Amer. J. Math. 56 (1934), no. 1-4, 87–95. MR 1507004, DOI 10.2307/2370916
Morgan Ward, Questions, Discussions, and Notes: On the Vanishing of the Sum of the $N$th Powers of the Roots of a Cubic Equation, Amer. Math. Monthly 41 (1934), no. 5, 313–316. MR 1523091, DOI 10.2307/2300994
Morgan Ward, Note on the period of a mark in a finite field, Bull. Amer. Math. Soc. 40 (1934), no. 4, 279–281. MR 1562836, DOI 10.1090/S0002-9904-1934-05847-6

Note on an arithmetical property of recurring series

39

Morgan Ward, An arithmetical property of recurring series of the second order, Bull. Amer. Math. Soc. 40 (1934), no. 12, 825–828. MR 1562981, DOI 10.1090/S0002-9904-1934-05972-X
Morgan Ward, Note on the iteration of functions of one variable, Bull. Amer. Math. Soc. 40 (1934), no. 10, 688–690. MR 1562953, DOI 10.1090/S0002-9904-1934-05942-1

The numerical evaluation of a class of trigonometric series

41

Morgan Ward, An enumerative problem in the arithmetic of linear recurring series, Trans. Amer. Math. Soc. 37 (1935), no. 3, 435–440. MR 1501795, DOI 10.1090/S0002-9947-1935-1501795-0
Morgan Ward, A Determination of all Possible Systems of Strict Implication, Amer. J. Math. 57 (1935), no. 2, 261–266. MR 1507068, DOI 10.2307/2371202

Conditions for factorization in a set closed under a single operation

36

Morgan Ward, On the factorization of polynomials to a prime modulus, Ann. of Math. (2) 36 (1935), no. 4, 870–874. MR 1503257, DOI 10.2307/1968592

The diophantine equation

38

Morgan Ward, A Calculus of Sequences, Amer. J. Math. 58 (1936), no. 2, 255–266. MR 1507149, DOI 10.2307/2371035
Morgan Ward and F. B. Fuller, The continuous iteration of real functions, Bull. Amer. Math. Soc. 42 (1936), no. 6, 393–396. MR 1563308, DOI 10.1090/S0002-9904-1936-06311-1
Morgan Ward, The null divisors of linear recurring series, Duke Math. J. 2 (1936), no. 3, 472–476. MR 1545940, DOI 10.1215/S0012-7094-36-00240-5
Morgan Ward, Note on divisibility sequences, Bull. Amer. Math. Soc. 42 (1936), no. 12, 843–845. MR 1563448, DOI 10.1090/S0002-9904-1936-06435-9
Morgan Ward, Linear divisibility sequences, Trans. Amer. Math. Soc. 41 (1937), no. 2, 276–286. MR 1501902, DOI 10.1090/S0002-9947-1937-1501902-1
Morgan Ward, Arithmetic functions on rings, Ann. of Math. (2) 38 (1937), no. 3, 725–732. MR 1503363, DOI 10.2307/1968611
Morgan Ward, Some Arithmetical Applications of Residuation, Amer. J. Math. 59 (1937), no. 4, 921–926. MR 1507292, DOI 10.2307/2371358
Morgan Ward, Residuation in structures over which a multiplication is defined, Duke Math. J. 3 (1937), no. 4, 627–636. MR 1546017, DOI 10.1215/S0012-7094-37-00351-X
Morgan Ward, Arithmetical properties of sequences in rings, Ann. of Math. (2) 39 (1938), no. 1, 210–219. MR 1503399, DOI 10.2307/1968724

Residuated lattices

24

Morgan Ward, The law of apparition of primes in a Lucasian sequence, Trans. Amer. Math. Soc. 44 (1938), no. 1, 68–86. MR 1501963, DOI 10.1090/S0002-9947-1938-1501963-0
Morgan Ward, Structure residuation, Ann. of Math. (2) 39 (1938), no. 3, 558–568. MR 1503424, DOI 10.2307/1968634
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc. 45 (1939), no. 4, 334–336. MR 1563977, DOI 10.1090/S0002-9904-1939-06980-2
Morgan Ward and R. P. Dilworth, Evaluations over residuated structures, Ann. of Math. (2) 40 (1939), no. 2, 328–338. MR 1503460, DOI 10.2307/1968921
Morgan Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939), no. 3, 335–354. MR 1501995, DOI 10.1090/S0002-9947-1939-1501995-3
Morgan Ward, The algebra of lattice functions, Duke Math. J. 5 (1939), no. 2, 357–371. MR 1546130, DOI 10.1215/S0012-7094-39-00531-4
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Morgan Ward, A characterization of Dedekind structures, Bull. Amer. Math. Soc. 45 (1939), no. 6, 448–451. MR 1564002, DOI 10.1090/S0002-9904-1939-07005-5
Morgan Ward, Note on the general rational solution of the equation $ax^2-by^2=z^3$, Amer. J. Math. 61 (1939), 788–790. MR 23, DOI 10.2307/2371337
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Morgan Ward, Residuated distributive lattices, Duke Math. J. 6 (1940), 641–651. MR 2845

The arithmetical properties of modular lattices

430

Morgan Ward, The closure operators of a lattice, Ann. of Math. (2) 43 (1942), 191–196. MR 6144, DOI 10.2307/1968865
Morgan Ward, Euler’s three biquadrate problem, Proc. Nat. Acad. Sci. U.S.A. 31 (1945), 125–127. MR 12104, DOI 10.1073/pnas.31.4.125
Morgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31–74. MR 23275, DOI 10.2307/2371930
Morgan Ward, Euler’s problem on sums of three fourth powers, Duke Math. J. 15 (1948), 827–837. MR 27287
Morgan Ward, The law of repetition of primes in an elliptic divisibility sequence, Duke Math. J. 15 (1948), 941–946. MR 27286
Morgan Ward, Mathematical Notes: A Generalized Integral Test for Convergence of Series, Amer. Math. Monthly 56 (1949), no. 3, 170–172. MR 1527199, DOI 10.2307/2305031
R. P. Dilworth and Morgan Ward, Note on a paper by C. E. Rickart, Bull. Amer. Math. Soc. 55 (1949), 1141. MR 32579, DOI 10.1090/S0002-9904-1949-09342-4
Morgan Ward, Arithmetical properties of the elliptic polynomials arising from the real multiplication of the Jacobi functions, Amer. J. Math. 72 (1950), 284–302. MR 33857, DOI 10.2307/2372033
Morgan Ward, Arithmetical properties of polynomials associated with the lemniscate elliptic functions, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 359–362. MR 36776, DOI 10.1073/pnas.36.6.359
Morgan Ward, A class of soluble Diophantine equations, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 113–114. MR 41870, DOI 10.1073/pnas.37.2.113
Morgan Ward, Prime divisors of second order recurring sequences, Duke Math. J. 21 (1954), 607–614. MR 64073
Morgan Ward, The maximal prime divisors of linear recurrences, Canadian J. Math. 6 (1954), 455–462. MR 66408, DOI 10.4153/CJM-1954-047-9
Morgan Ward, Cyclotomy and the converse of Fermat’s theorem, Amer. Math. Monthly 61 (1954), 564. MR 63397, DOI 10.2307/2307608
Morgan Ward, The intrinsic divisors of Lehmer numbers, Ann. of Math. (2) 62 (1955), 230–236. MR 71446, DOI 10.2307/1969677
Morgan Ward, On the number of vanishing terms in an integral cubic recurrence, Amer. Math. Monthly 62 (1955), 155–160. MR 67136, DOI 10.2307/2306618
Morgan Ward, The laws of apparition and repetition of primes in a cubic recurrence, Trans. Amer. Math. Soc. 79 (1955), 72–90. MR 68579, DOI 10.1090/S0002-9947-1955-0068579-7
Morgan Ward, The mappings of the positive integers into themselves which preserve division, Pacific J. Math. 5 (1955), 1013–1023. MR 75968
Morgan Ward, Tests for primality based on Sylvester’s cyclotomic numbers, Pacific J. Math. 9 (1959), 1269–1272. MR 108464
Morgan Ward, The vanishing of the homogeneous product sum of the roots of a cubic, Duke Math. J. 26 (1959), 553–562. MR 110669

Some diophantine problems connected with linear recurrences

Morgan Ward, The vanishing of the homogeneous product sum on three letters, Duke Math. J. 27 (1960), 619–624. MR 114790
Morgan Ward, The calculation of the complete elliptic integral of the third kind, Amer. Math. Monthly 67 (1960), 205–213. MR 136053, DOI 10.2307/2309677
Morgan Ward, The prime divisors of Fibonacci numbers, Pacific J. Math. 11 (1961), 379–386. MR 138586
Morgan Ward, The linear $p$-adic recurrence of order two, Illinois J. Math. 6 (1962), 40–52. MR 138585
E. C. Dade, D. W. Robinson, O. Taussky, and M. Ward, Divisors of recurrent sequences, J. Reine Angew. Math. 214(215) (1964), 180–183. MR 161875, DOI 10.1515/crll.1964.214-215.180