To give an indication of his father's work during the time that Dick was growing up, let us mention that DNL published Factor table for the first ten millions when Dick was four years old, and List of prime numbers from 1 to 10006721 when he was nine. Dick was fascinated to listen to his father describe mathematical ideas to him even at this age. After completing his school education, Dick entered Berkeley to study physics. He was now highly involved with his father's ideas so on the one hand he studied physics courses, while on the other hand he helped his father both with the number theory computations he was undertaking and with the mechanical ideas that he was developing to help him make these calculations.
One project that DNL worked on during the time that Dick was an undergraduate was his work on Factor Stencils which was published in 1929. This gave a method of factorising a number using cards with holes punched in them and it was later described by Dick Lehmer as follows:-
Since every quadratic residue R of a number N is also a quadratic residue of every possible factor of N, it follows that the problem of factoring a number N is hereby reduced to the discovery of an adequate number of quadratic residues R of N and the superposition of the corresponding stencils to reveal those few primes having these residues R.While he was a physics undergraduate, Dick helped his father to produce the stencils. They worked with residues R < 240 and covered 5000 primes which includes all primes up to 48611. This enabled integers up to 486112 = 2363029321 to be factored. While DHL and his father were working on the stencils another undergraduate at Berkeley, Emma Trotskaia who was a mathematics student studying DNL's courses, assisted them. Emma would soon become Dick's wife and Dick and Emma Lehmer would become one of the most famous husband and wife mathematics teams.
In 1927 Lehmer graduated from Berkeley with a B.A. degree in Physics and he went to the University of Chicago to undertake research for his doctorate in mathematics with L E Dickson as his supervisor. In the following year Emma Trotskaia graduated with a B.A. degree with honours in Mathematics and, just prior to her taking her final examinations, Dick and Emma Lehmer were married. Once Emma's exams were over, they set out on a tour which began in the redwood forest, went on to Japan, and finally a visit which let Dick meet his new wife's family before they returned to Berkeley. Dick was not happy working under Dickson in Chicago so he had accepted an offer of an instructorship at Brown University in Providence, Rhode Island. The recently married couple drove across the United States to Brown University where both enrolled for a Master's degree.
Lehmer was awarded his Master's Degree in 1929 and his doctorate, also from Brown University, in 1930. His dissertation, which was supervised by Tamarkin, was An Extended Theory of Lucas's Functions. Lehmer's life over the next few years involved moving from place to place hoping for a permanent university post in the particularly difficult times of the Great Depression. After receiving his doctorate, Lehmer was awarded a National Research Fellowship and with this he spent 1930-31 at the California Institute of Technology and then 1931-32 at Stanford. After a spell at the Institute for Advanced Study at Princeton, where he held a second Fellowship, Lehmer moved to a more permanent post at Lehigh University in Pennsylvania.
Lehmer and his wife remained at Lehigh until 1940 except for the year 1938-39 which they spent in England visiting both the University of Cambridge and the University of Manchester. In England they met, among others, Hardy, Littlewood, Davenport, Mahler, Mordell, and Erdős. Back in the United States not long after the outbreak of World War II, Lehmer spent another year at Lehigh before accepting a post in Berkeley in 1940. It was the job he always wanted and it was a great joy to Lehmer and his family to return home.
The Lehmers spent 1945-46 at the Aberdeen Proving Ground where Lehmer's task was to help set up and operate the ENIAC (Electronic Numerical Integrator and Calculator) computer. Although the computer worked most of the time computing trajectories for ballistics problems, on some weekends the Lehmers used it to solve certain number theory problems using it as an electronic sieve :-
When they could arrange child care, they often stayed at the lab all night long while the ENIAC processed one of their problems. They would return home at the break of dawn. They were pleased to find that the sieve worked in successfully solving problems.In February 1950 Senator Joseph R McCarthy of Wisconsin claimed that 205 State Department employees were communists who were disloyal to the United States. McCarthy enjoyed a highly successful few years by making these charges of disloyalty that, though mostly undocumented, badly hurt government employees, teachers, and university professors. Although McCarthy was the most prominent person taking this line, it was a road that the United States was already on and the State of California had for some time been discussing loyalty oaths. In 1950 the Board of Regents of the State of California decided to implement a policy that all employees sign a loyalty oath, and the University of California at Berkeley was chosen as one of the first test cases for it. Nineteen faculty members of the University of California refused; Lehmer was one such faculty member.
The assumption was that anyone who would not sign the oath must be a communist sympathiser and must be sacked, so Lehmer, as one of those who refused, lost his position. Many considered that the oath violated their rights of academic freedom which university researchers valued most highly. Of course the witch-hunts against imaginary communists in the early 1950s was a disgraceful affair which cost many their jobs and led to long-term suffering. For Lehmer, however, the problem was not so acute for he was able to take up the post of Director of the National Bureau of Standards' Institute for Numerical Analysis for the time that he was unable to hold his faculty position in Berkeley. Others were not as fortunate as he was and suffered real hardships. After the courts proclaimed the oath to be unconstitutional, Lehmer was reinstated at Berkeley.
Lehmer's Selected Papers published in 1981 gives a good indication of the range of topics on which he worked. The chapter headings are: Lucas's functions; Tests for primality; Continued fractions; Bernoulli numbers and polynomials; Diophantine equations; Numerical functions; Matrices; Power residues; Analytic number theory; Partitions; Modular forms; Cyclotomy; Combinatorics; Sieves; Equation solving; Computing techniques; and Miscellaneous. His most famous monograph was Guide to Tables in the Theory of Numbers. R D Carmichael, reviewing the book, wrote:-
A descriptive account is given of existing tables in the theory of numbers; this is set forth in such a way as to indicate clearly what each table contains. A bibliography, arranged alphabetically by authors, gives exact references to the material cited and supplies information concerning the holdings, in libraries of the United States and Canada, of the books and pamphlets to which reference is made. Errata in the tables are listed, the sources being given in the cases of errata previously printed; Lehmer's contributions in the way of new indications of errata are notable.Lehmer was awarded an honorary degree from Brown University in 1980. The citation reads in part:-
Prolific in research, you have made far-reaching contributions to number theory. You were among the first to recognize the importance of high-speed computers as an aid to mathematical research. With great energy and enthusiasm, you demonstrated how, in both theory and practice, computers can be an invaluable tool in testing conjectures.Lehmer lectured at the International Conference on Computers and Mathematics held at Stanford University in 1986. His talk, Factorization then and now, covered one of the topics to which he had made major contributions. He was a pioneer in the application of mechanical methods, including digital computers, to the solution of problems in number theory and he talked about some of the methods used to factorise numbers including: factor tables, trial division, Legendre's method, factor stencils, the continued fraction method, Fermat's method, methods based on quadratic forms, and Shanks' method.
Let us mention a number of other topics for which Lehmer will be remembered. One must be the Lucas-Lehmer primality test which uses the Fermat congruence, and in particular his application to testing whether a Mersenne number was prime. He also made major contributions to studying the density of primes with a given primitive root and to the study of the partition function, in particular verifying certain conjectures by Ramanujan. He was the first person to attack the Riemann Hypothesis by using a computer to check if the roots lie on the critical line. Luck, however, often plays a large role in determining how famous a mathematician will become, and Lehmer's attack on the Riemann Hypothesis only provided evidence that the hypothesis was true whereas had the world been different it might have yielded a counterexample.
Brillhart, who received a Ph.D. in 1967 for a thesis supervised by Lehmer, comments in  on Lehmer as a lecturer:-
As a lecturer [Lehmer] was much appreciated not only for his classical scholarship in mathematics and number theory, but also for his dry sense of humour and wit.Brillhart also comments:-
As a thinker [Lehmer] was sagaciously independent, not being devoted to dogmas, systems, or rituals.
Article by: J J O'Connor and E F Robertson