The mathematical work of Morgan Ward

Author:
D. H. Lehmer

Journal:
Math. Comp. **61** (1993), 307-311

MSC:
Primary 01A70; Secondary 11-03

DOI:
https://doi.org/10.1090/S0025-5718-1993-1182245-3

MathSciNet review:
1182245

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**[1]**Morgan Ward,*A generalization of recurrents*, Bull. Amer. Math. Soc.**33**(1927), no. 4, 477–492. MR**1561403**, https://doi.org/10.1090/S0002-9904-1927-04410-X**[2]***General arithmetic*, Proc. Nat. Acad. Sci. U.S.A.**13**(1927), 748-749.**[3]***Postulates for an abstract arithmetic*, Proc. Nat. Acad. Sci. U.S.A.**14**(1928), 907-911.**[4]**Morgan Ward,*A Simplification of Certain Problems in Arithmetical Division*, Amer. Math. Monthly**35**(1928), no. 1, 9–14. MR**1521339**, https://doi.org/10.2307/2298578**[5]**Morgan Ward,*On Certain Functional Relations*, Amer. Math. Monthly**36**(1929), no. 8, 426–431. MR**1521822**, https://doi.org/10.2307/2299940**[6]***Certain expansions involving doubly infinite series*, Ann. of Math. (2)**30**(1929), 578-582.**[7]**Morgan Ward,*A certain class of polynomials*, Ann. of Math. (2)**31**(1930), no. 1, 43–51. MR**1502917**, https://doi.org/10.2307/1968138**[8]***A generalization of a familiar theorem concerning prime numbers*, J. London Math. Soc.**5**(1930), 106-107.**[9]***The reversion of a power series*, Rend. Circ. Mat. Palermo**54**(1930), 42-46.**[10]**Morgan Ward,*Postulates for the inverse operations in a group*, Trans. Amer. Math. Soc.**32**(1930), no. 3, 520–526. MR**1501550**, https://doi.org/10.1090/S0002-9947-1930-1501550-7**[11]**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**, https://doi.org/10.1090/S0002-9947-1931-1501582-X**[12]**Morgan Ward,*The algebra of recurring series*, Ann. of Math. (2)**32**(1931), no. 1, 1–9. MR**1502975**, https://doi.org/10.2307/1968408**[13]**Morgan Ward,*Some arithmetical properties of sequences satisfying a linear recursion relation*, Ann. of Math. (2)**32**(1931), no. 4, 734–738. MR**1503026**, https://doi.org/10.2307/1968316**[14]**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**, https://doi.org/10.1090/S0002-9947-1931-1501583-1**[15]**Morgan Ward,*Conditions for the solubility of the Diophantine equation 𝑥²-𝑀𝑦²=-1*, Trans. Amer. Math. Soc.**33**(1931), no. 3, 712–718. MR**1501611**, https://doi.org/10.1090/S0002-9947-1931-1501611-3**[16]**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**, https://doi.org/10.2307/1968334**[17]***On the behaviour of non-static models of the universe when the cosmological term is omitted*, (with R. C. Tolman), Phys. Rev.**39**(1932), 835-843.**[18]**Morgan Ward,*A Type of Multiplicative Diophantine System*, Amer. J. Math.**55**(1933), no. 1-4, 67–76. MR**1506944**, https://doi.org/10.2307/2371110**[19]***A property of recurring series*, Proc. Nat. Acad. Sci. U.S.A.**19**(1933), 914-916.**[20]**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**, https://doi.org/10.1090/S0002-9947-1933-1501682-6**[21]**Morgan Ward,*The arithmetical theory of linear recurring series*, Trans. Amer. Math. Soc.**35**(1933), no. 3, 600–628. MR**1501705**, https://doi.org/10.1090/S0002-9947-1933-1501705-4**[22]**Morgan Ward,*Questions, Discussions, and Notes: A Certain Class of Trigonometric Integrals*, Amer. Math. Monthly**40**(1933), no. 6, 340–346. MR**1522833**, https://doi.org/10.2307/2301523**[23]**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**, https://doi.org/10.2307/2370916**[24]**Morgan Ward,*Questions, Discussions, and Notes: On the Vanishing of the Sum of the 𝑁th Powers of the Roots of a Cubic Equation*, Amer. Math. Monthly**41**(1934), no. 5, 313–316. MR**1523091**, https://doi.org/10.2307/2300994**[25]**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**, https://doi.org/10.1090/S0002-9904-1934-05847-6**[26]***Note on an arithmetical property of recurring series*, Math. Z.**39**(1934), 211-214.**[27]**Morgan Ward,*An arithmetical property of recurring series of the second order*, Bull. Amer. Math. Soc.**40**(1934), no. 12, 825–828. MR**1562981**, https://doi.org/10.1090/S0002-9904-1934-05972-X**[28]**Morgan Ward,*Note on the iteration of functions of one variable*, Bull. Amer. Math. Soc.**40**(1934), no. 10, 688–690. MR**1562953**, https://doi.org/10.1090/S0002-9904-1934-05942-1**[29]***The numerical evaluation of a class of trigonometric series*, Amer. Math. Monthly**41**(1934), 563-565.**[30]**Morgan Ward,*An enumerative problem in the arithmetic of linear recurring series*, Trans. Amer. Math. Soc.**37**(1935), no. 3, 435–440. MR**1501795**, https://doi.org/10.1090/S0002-9947-1935-1501795-0**[31]**Morgan Ward,*A Determination of all Possible Systems of Strict Implication*, Amer. J. Math.**57**(1935), no. 2, 261–266. MR**1507068**, https://doi.org/10.2307/2371202**[32]***Conditions for factorization in a set closed under a single operation*, Ann. of Math. (2)**36**(1935), 36-39.**[33]**Morgan Ward,*On the factorization of polynomials to a prime modulus*, Ann. of Math. (2)**36**(1935), no. 4, 870–874. MR**1503257**, https://doi.org/10.2307/1968592**[34]***The diophantine equation*, Trans. Amer. Math. Soc.**38**(1935), 447-457.**[35]**Morgan Ward,*A Calculus of Sequences*, Amer. J. Math.**58**(1936), no. 2, 255–266. MR**1507149**, https://doi.org/10.2307/2371035**[36]**Morgan Ward and F. B. Fuller,*The continuous iteration of real functions*, Bull. Amer. Math. Soc.**42**(1936), no. 6, 393–396. MR**1563308**, https://doi.org/10.1090/S0002-9904-1936-06311-1**[37]**Morgan Ward,*The null divisors of linear recurring series*, Duke Math. J.**2**(1936), no. 3, 472–476. MR**1545940**, https://doi.org/10.1215/S0012-7094-36-00240-5**[38]**Morgan Ward,*Note on divisibility sequences*, Bull. Amer. Math. Soc.**42**(1936), no. 12, 843–845. MR**1563448**, https://doi.org/10.1090/S0002-9904-1936-06435-9**[39]**Morgan Ward,*Linear divisibility sequences*, Trans. Amer. Math. Soc.**41**(1937), no. 2, 276–286. MR**1501902**, https://doi.org/10.1090/S0002-9947-1937-1501902-1**[40]**Morgan Ward,*Arithmetic functions on rings*, Ann. of Math. (2)**38**(1937), no. 3, 725–732. MR**1503363**, https://doi.org/10.2307/1968611**[41]**Morgan Ward,*Some Arithmetical Applications of Residuation*, Amer. J. Math.**59**(1937), no. 4, 921–926. MR**1507292**, https://doi.org/10.2307/2371358**[42]**Morgan Ward,*Residuation in structures over which a multiplication is defined*, Duke Math. J.**3**(1937), no. 4, 627–636. MR**1546017**, https://doi.org/10.1215/S0012-7094-37-00351-X**[43]**Morgan Ward,*Arithmetical properties of sequences in rings*, Ann. of Math. (2)**39**(1938), no. 1, 210–219. MR**1503399**, https://doi.org/10.2307/1968724**[44]***Residuated lattices*, Proc. Nat. Acad. Sci. U.S.A.**24**(1938), 162-164.**[45]**Morgan Ward,*The law of apparition of primes in a Lucasian sequence*, Trans. Amer. Math. Soc.**44**(1938), no. 1, 68–86. MR**1501963**, https://doi.org/10.1090/S0002-9947-1938-1501963-0**[46]**Morgan Ward,*Structure residuation*, Ann. of Math. (2)**39**(1938), no. 3, 558–568. MR**1503424**, https://doi.org/10.2307/1968634**[47]**Morgan Ward,*A note on divisibility sequences*, Bull. Amer. Math. Soc.**45**(1939), no. 4, 334–336. MR**1563977**, https://doi.org/10.1090/S0002-9904-1939-06980-2**[48]**Morgan Ward and R. P. Dilworth,*Evaluations over residuated structures*, Ann. of Math. (2)**40**(1939), no. 2, 328–338. MR**1503460**, https://doi.org/10.2307/1968921**[49]**Morgan Ward and R. P. Dilworth,*Residuated lattices*, Trans. Amer. Math. Soc.**45**(1939), no. 3, 335–354. MR**1501995**, https://doi.org/10.1090/S0002-9947-1939-1501995-3**[50]**Morgan Ward,*The algebra of lattice functions*, Duke Math. J.**5**(1939), no. 2, 357–371. MR**1546130**, https://doi.org/10.1215/S0012-7094-39-00531-4**[51]**Morgan Ward,*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783–787. MR**0000010**, https://doi.org/10.2307/2371336**[52]**Morgan Ward,*A characterization of Dedekind structures*, Bull. Amer. Math. Soc.**45**(1939), no. 6, 448–451. MR**1564002**, https://doi.org/10.1090/S0002-9904-1939-07005-5**[53]**Morgan Ward,*Note on the general rational solution of the equation 𝑎𝑥²-𝑏𝑦²=𝑧³*, Amer. J. Math.**61**(1939), 788–790. MR**0000023**, https://doi.org/10.2307/2371337**[54]**Morgan Ward and R. P. Dilworth,*The lattice theory of ova*, Ann. of Math. (2)**40**(1939), 600–608. MR**0000011**, https://doi.org/10.2307/1968944**[55]**Garrett Birkhoff and Morgan Ward,*A characterization of Boolean algebras*, Ann. of Math. (2)**40**(1939), 609–610. MR**0000009**, https://doi.org/10.2307/1968945**[56]**Morgan Ward,*Residuated distributive lattices*, Duke Math. J.**6**(1940), 641–651. MR**0002845****[57]***The arithmetical properties of modular lattices*, Rev. Ci. Lima**430**(1941), 593-603.**[58]**Morgan Ward,*The closure operators of a lattice*, Ann. of Math. (2)**43**(1942), 191–196. MR**0006144**, https://doi.org/10.2307/1968865**[59]**Morgan Ward,*Euler’s three biquadrate problem*, Proc. Nat. Acad. Sci. U. S. A.**31**(1945), 125–127. MR**0012104****[60]**Morgan Ward,*Memoir on elliptic divisibility sequences*, Amer. J. Math.**70**(1948), 31–74. MR**0023275**, https://doi.org/10.2307/2371930**[61]**Morgan Ward,*Euler’s problem on sums of three fourth powers*, Duke Math. J.**15**(1948), 827–837. MR**0027287****[62]**Morgan Ward,*The law of repetition of primes in an elliptic divisibility sequence*, Duke Math. J.**15**(1948), 941–946. MR**0027286****[63]**Morgan Ward,*Mathematical Notes: A Generalized Integral Test for Convergence of Series*, Amer. Math. Monthly**56**(1949), no. 3, 170–172. MR**1527199**, https://doi.org/10.2307/2305031**[64]**R. P. Dilworth and Morgan Ward,*Note on a paper by C. E. Rickart*, Bull. Amer. Math. Soc.**55**(1949), 1141. MR**0032579**, https://doi.org/10.1090/S0002-9904-1949-09342-4**[65]**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**0033857**, https://doi.org/10.2307/2372033**[66]**Morgan Ward,*Arithmetical properties of polynomials associated with the lemniscate elliptic functions*, Proc. Nat. Acad. Sci. U. S. A.**36**(1950), 359–362. MR**0036776****[67]**Morgan Ward,*A class of soluble Diophantine equations*, Proc. Nat. Acad. Sci. U. S. A.**37**(1951), 113–114. MR**0041870****[68]**Morgan Ward,*Prime divisors of second order recurring sequences*, Duke Math. J.**21**(1954), 607–614. MR**0064073****[69]**Morgan Ward,*The maximal prime divisors of linear recurrences*, Canad. J. Math.**6**(1954), 455–462. MR**0066408**, https://doi.org/10.4153/CJM-1954-047-9**[70]**Morgan Ward,*Cyclotomy and the converse of Fermat’s theorem*, Amer. Math. Monthly**61**(1954), 564. MR**0063397**, https://doi.org/10.2307/2307608**[71]**Morgan Ward,*The intrinsic divisors of Lehmer numbers*, Ann. of Math. (2)**62**(1955), 230–236. MR**0071446**, https://doi.org/10.2307/1969677**[72]**Morgan Ward,*On the number of vanishing terms in an integral cubic recurrence*, Amer. Math. Monthly**62**(1955), 155–160. MR**0067136**, https://doi.org/10.2307/2306618**[73]**Morgan Ward,*The laws of apparition and repetition of primes in a cubic recurrence*, Trans. Amer. Math. Soc.**79**(1955), 72–90. MR**0068579**, https://doi.org/10.1090/S0002-9947-1955-0068579-7**[74]**Morgan Ward,*The mappings of the positive integers into themselves which preserve division*, Pacific J. Math.**5**(1955), 1013–1023. MR**0075968****[75]**Morgan Ward,*Tests for primality based on Sylvester’s cyclotomic numbers*, Pacific J. Math.**9**(1959), 1269–1272. MR**0108464****[76]**Morgan Ward,*The vanishing of the homogeneous product sum of the roots of a cubic.*, Duke Math. J**26**(1959), 553–562. MR**0110669****[77]***Some diophantine problems connected with linear recurrences*, Report of the Institute in the Theory of Numbers, Univ. Colorado, Boulder, Colorado, 1959, pp. 250-257.**[78]**Morgan Ward,*The vanishing of the homogeneous product sum on three letters*, Duke Math. J.**27**(1960), 619–624. MR**0114790****[79]**Morgan Ward,*The calculation of the complete elliptic integral of the third kind.*, Amer. Math. Monthly**67**(1960), 205–213. MR**0136053**, https://doi.org/10.2307/2309677**[80]**Morgan Ward,*The prime divisors of Fibonacci numbers*, Pacific J. Math.**11**(1961), 379–386. MR**0138586****[81]**Morgan Ward,*The linear 𝑝-adic recurrence of order two*, Illinois J. Math.**6**(1962), 40–52. MR**0138585****[82]**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**0161875**

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DOI:
https://doi.org/10.1090/S0025-5718-1993-1182245-3

Article copyright:
© Copyright 1993
American Mathematical Society