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Daniel Shanks 1917-1996 comm-shanks.qxp 3/24/98 9:49 AM Page 813 Daniel Shanks (1917–1996) H. C. Williams Daniel Shanks was born on January 17, 1917, in versity (as all the city of Chicago, where he was raised and universities will) where in 1937 he received his B.S. in physics from insisted that he the University of Chicago. In 1940 he worked as complete all a physicist at the Aberdeen Proving Grounds, their degree re- moving the following year to the position of quirements be- physicist at the Naval Ordinance Laboratory, a fore being position he retained until 1950. In 1951 his post awarded the de- at the NOL changed to that of mathematician, gree. At the time and during the years from 1951 to 1957 he Dan was raising headed the Numerical Analysis Section and then a young family the Applied Mathematics Laboratory. He left the and working full NOL in 1957 to become consultant and senior time, so it was research scientist in the Computation and Math- not until 1954 ematics Department at the Naval Ship R&D Cen- that he obtained ter at the David Taylor Model Basin. In 1976 his degree. His after support for independent work had con- thesis was pub- siderably diminished, he decided to retire, spend- lished in 1955 in ing a year as a guest worker at the National Bu- the Journal of reau of Standards. He joined the Department of Mathematics and Mathematics at the University of Maryland as an Physics and was adjunct professor in 1977 and remained there entitled “Non- until his death on September 6, 1996. He is sur- linear Transformations of Divergent and Slowly vived by two sisters; his sons, Leonard and Oliver; Convergent Sequences”. It concerned methods an adopted son, Gabriel; and two grandchildren. of accelerating the convergence of slowly con- Dan (he insisted that everyone call him Dan) vergent sequences and is now considered a clas- received his Ph.D. from the University of Mary- sic in its field. The transformation that he in- land in 1954, but it was as early as 1949, before troduced is today referred to as the Shanks having done any graduate work, that he pre- transformation. Dan considered this paper to be sented his thesis to the somewhat surprised De- one of his two most important published works. partment of Mathematics. It was at this point that Dan served as an editor of Mathematics of he requested a Ph.D. in mathematics should the Computation from 1959 until his death. Through- work be judged of sufficient quality. There was out almost all of this period he was extremely no question concerning the excellence of the active in all aspects of the journal’s operation work—indeed, the final thesis was little differ- through his efforts in publishing his own work, ent from the original submission—but the Uni- soliciting papers that he regarded as being of par- Hugh Williams is a professor in the Department of ticular significance, encouraging young math- Computer Science at the University of Manitoba, Win- ematicians in their researches, reviewing tables, nipeg MB, Canada. His e-mail address is copyediting and even, when occasion demanded, [email protected]. serving unofficially in the capacity of managing AUGUST 1997 NOTICES OF THE AMS 813 comm-shanks.qxp 3/24/98 9:49 AM Page 814 editor. construction of tables and are frequently in- Many of spired from the examination of tables. his re- His early work (1951–58) was mainly devoted views of to numerical analytic topics, interests which led tables in 1962 to his most famous paper. This is his were work with John Wrench on the computation of suffi- π to 100000 decimals, a considerable improve- ciently ment over all previous work on this problem. Sev- Figure 1. lengthy, eral new ideas for effecting such large-scale com- detailed, putations were mentioned in this paper, but Dan and in- never seemed to have regarded it as highly as sightful to be research papers in their own right. several of his later works. It should also be men- He also served as the custodian of the UMT (Un- tioned that in 1962 he published the first edi- published Mathematical Tables) file. Fortunately, tion of his book Solved and Unsolved Problems this file, while no longer maintained by Math- in Number Theory. This is a charming, uncon- ematics of Computation, is temporarily being ventional, provocative, and fascinating book on preserved for archival purposes through the generous efforts of Duncan Buell of the Center elementary number theory which has seen three for Computing Sciences in Bowie, MD. It is hoped further editions, the latest in 1993. It is the sec- that a permanent home for the UMT will be ond of his works of which Dan was most proud. found soon. In 1987 Dan was honored by the Dan’s earlier investigations in number theory publication of a special issue of Mathematics of began with his interest in the distribution of Computation dedicated to him on the occasion primes. In particular, he studied primes of the of his seventieth birthday and commemorating form n2 + a, n4 +1, and n6 + 1091. In the last his many contributions to computational num- case he found that there was only one prime of ber theory. For it was in this area of research and the form n6 + 1091 for 1 n 4000 and gave ≤ ≤ related journal activities that he made his great- a heuristic explanation as to why one would not est contribution to M of C, a periodical which has expect to find many primes of this form. He de- now become the journal of choice for publica- veloped a sieve technique to search for primes tions in both numerical analysis and computa- of the form n2 +1; a refinement of the same idea tional number theory. This rather odd combi- was used later by Carl Pomerance in his very suc- nation of areas was initiated through D. H. cessful quadratic sieve method for factoring in- Lehmer’s association with M of C at its very be- tegers. ginning in 1943, when it was called Mathemati- If we let Pa(N) represent the number of cal Tables and Other Aids to Computation primes of the form n2 + a for 1 n N and (MTAC). That the number theoretic component ≤ ≤ π a(N) denote the number of primes N for ≤ of M of C has continued to flourish to this day which a is a quadratic nonresidue, then a con- − is in no small part due to Dan’s tireless efforts jecture of Hardy and Littlewood in 1923 implies and his relentless insistence on the quality of the that articles that appear in this journal. Pa(N) ( ) ha, Dan wrote over eighty papers and one book. ∗ π a(N) ∼ The main areas to which he made contributions were: numerical analysis, distribution of primes, Dirichlet series, quadratic forms (fields), class where ha is given by the very slowly convergent group invariants, and computational algorithms infinite product over the primes p in number fields of degree 3 and 4. He wrote as a 1 well on a variety of other topics, such as: black ha = 1 ( − ) − p p 1 body radiation, ballistics, mathematical identi- pYa − ties, Epstein zeta functions, formulas for π, and \ primality testing by means of cubic recurrences. and ( /p) is the Legendre symbol. Dan gave a · Dan’s papers are most frequently characterized heuristic argument in support of (*), ran a num- by an experimental approach to their subject ber of numerical experiments to verify (*) for N matter, likely as a result of his training as a up to 180000, and developed a fast method to physicist. evaluate ha. Indeed, he computed ha for a =1, He illustrated his philosophy concerning this 2, 3, 4, 5, 6, 7 to 8 decimals and gave ± ± ± ± ± in a survey paper on algebraic number fields as 3 decimal estimates of several others. shown in Figure 1. Here the arrows signify the It was this study that led him to his work on connection between the various topics. For ex- Dirichlet series and class number computation ample, “ideas and problems”, which are of the for quadratic forms. For a given character χ the greatest interest mathematically, motivate the Dirichlet series is defined by 814 NOTICES OF THE AMS VOLUME 44, NUMBER 7 comm-shanks.qxp 3/24/98 9:49 AM Page 815 this technique can now evaluate R for values of ∞ s L(s,χ)= χ(n)n− . d between 40 and 60 digits. nX=1 Dan turned these investigations toward the If χ(n) is the Kronecker symbol (d/n), then problem of factoring integers. As he knew that the discovery of an ambiguous form of dis- he showed that ha could be given by formulas criminant d could be used to factor d, it was sim- like ply a matter of finding elements of the 2-sylow subgroup of the class group of quadratic forms h L(1,χ)L(2,χ) a of discriminant d(< 0). His work resulted in a (4) means of factoring d in O( d 1/4+ε) operations, | | 1 1 2 a very good result at the time. He also showed = 1 1 , 2 − p4 − q(q 1)2 how to use the theory of continued fractions to p a q 3 Y| Y≥ − search for reduced ambiguous forms of positive discriminant d. He called this technique SQUFOF where d = a and the products are evaluated (square form factorization); it was very simple − over the primes.
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