Derrick Henry Lehmer

Derrick Lehmer (1984)

Derrick Henry Lehmer (born February 23, 1905 in Berkeley (California) , † May 22, 1991 ibid) was an American mathematician specializing in number theory .

Life

As a child, Lehmer was drawn into his later field of work, as his father Derrick Norman Lehmer (1867–1938) created tables of prime numbers and factor tables with the help of mechanical calculators. He first studied physics at the University of California, Berkeley , where his father was a professor of mathematics. As a student he worked on the implementation of factoring algorithms for his father with punched card computers, with the support of his future wife Emma Markovna Trotskaya , who also studied at Berkeley. In 1927 he made his BA in physics and began a doctorate with the leading American number theorist Leonard Eugene Dickson in Chicagoto work. Since he could not cope with this, he moved to Jacob David Tamarkin at Brown University in Rhode Island , where he received his doctorate in 1930.

He then went to the California Institute of Technology with a state scholarship in 1930/31 and then went to Stanford University and the Institute for Advanced Study in Princeton (New Jersey) for a year before he was able to receive a lectureship at Lehigh University . Except for a visit to England in 1938/39 with Godfrey Harold Hardy , John Edensor Littlewood , Harold Davenport , Kurt Mahler , Louis Mordell and Paul Erdős , he and his wife stayed in Lehigh until 1940, before he was able to get a post at his home university, Berkeley. During the war years, he and his wife worked as operators for ENIAC on the US Army's Aberdeen test site : during the day on ballistic calculations, at night there was time for number theory. When he refused an oath of loyalty enforced by Joseph McCarthy in Berkeley in 1950 , he briefly lost his post, which he bridged with work for the National Bureau of Standards . After his reinstatement, he received recognition and honors, such as Vice President of the American Mathematical Society and Governor at Large of the Association for Computing Machinery 1953–1954.

When Raymond Clare Archibald founded the journal "Mathematical Tables and Other Aids to Computation" (MTAC, today "Mathematics of Computation") in January 1943 , he was a member of the advisory board. After Archibald retired in late 1949, Lehmer was First Chairman of the magazine's Editorial Committee from 1950 to 1954.

From 1954 to 1957 he headed the Mathematics Department at the University of California at Berkeley. In 1958 he was invited speaker at the International Congress of Mathematicians in Edinburgh ( Discrete variable methods in numerical analysis ). He retired in 1972. He received an honorary doctorate from Brown University in 1980.
Lehmer was born on the 50th anniversary of Carl Friedrich Gauß's death . This prompted Daniel Shanks to make a joking remark in 1989, referring to his joint work with Lehmer on a problem suggested by Gauss.

The Lehmers had a daughter (Laura Lehmer Gould, * 1932) and a son (Donald, * 1934).

plant

Lehmer was a pioneer in the use of computers or, in general, numerical methods in number theory. He found an improved version of of Édouard Lucas originating Lucas tests and other methods for detecting the primality of natural numbers. The Lucas-Lehmer test for Mersenne prime numbers is named after him and Lucas. He was also one of the first to electronically test the Riemann hypothesis . In doing so, he discovered unexpectedly closely spaced zeros of the Riemann zeta function , which are now called Lehmer pairs .

As a successor to his father, Lehmer also constructed various devices for the sieving process to calculate a solution of number-theoretical congruences, primarily for prime factorization - 1926 with bicycle chains, 1932 with optical gears (exhibited at the 1933/34 World Exhibition in Chicago), 1936 with 16 -mm film strips, 1966 Delay Line Sieve ("Dick Lehmers Sieve") DLS-127, 1969 DLS-157, 1975 Shift Register Sieve SRS-181, and programmed the sieving process on the computers SWAC, IBM 7094 and ILLIAC IV.

As early as 1938 he developed a much accelerated variant of the Euclidean algorithm for very large natural numbers. In 1959 he improved the astronomer Ernst Meissel's formula for the number of prime numbers up to a given limit and was able to calculate with his method . Shortly afterwards, however, its value turned out to be 1 too great. ${\displaystyle \pi (x)}$ ${\displaystyle x}$ ${\displaystyle \pi (10^{10})}$

The Lehmersche constant is that real number whose (first introduced by Lehmer) Kotangensentwicklung converges slowest. ${\displaystyle 0{,}5926327182...}$

Lehmer was concerned with the function originally examined by S. Ramanujan , defined by ${\displaystyle \tau }$

${\displaystyle \sum _{n=1}^{\infty }\tau (n)x^{n}=x((1-x)(1-x^{2})(1-x^{3})...)^{24}}$

and in 1947 set up Lehmer's conjecture that no natural value assumes the value zero (cf. Ramanujan's tau function ). ${\displaystyle \tau (n)}$ ${\displaystyle n}$

The Lehmer-Schur algorithm is a method for isolating the zeros of a polynomial in the complex plane. It is based on a criterion for determining the number of zeros of the polynomial in a circle with a given center point and radius and on the systematic generalization of the one-dimensional nesting of intervals.

A parametric method for averaging nonnegative numbers that provides some of the most common averages as special cases is called the Lehmer mean .

The (unsolved) Lehmer problem asks whether an integer polynomial 10th degree discovered by Lehmer with its zeros lying outside the complex unit circle can be exceeded with regard to its proximity to this unit circle (more precisely, after the existence of a lower bound for the so-called Mahler- Measure of a polynomial whose coefficients are whole numbers). The largest real zero of this polynomial is called Lehmer's number . ${\displaystyle C>1}$ ${\displaystyle 1{,}17628}$

The so-called totient problem by Lehmer is also one of the easy-to- ask , but so far completely unsolved questions : Is there a composite natural number such that Euler's φ-function applies? Such a number would be an extremely remarkable Fermatsche pseudoprime number , on the other hand the non-existence of such a number would result in the (still questionable) prime number criterion. ${\displaystyle n}$ ${\displaystyle \varphi (n)|(n-1)}$ ${\displaystyle n}$ ${\displaystyle \,n\equiv 1\mod \varphi (n)\,}$

As Lehmer matrices one of Lehmer given family is symmetric matrices with rational elements referred to. Their inverses are tridiagonal matrices with strictly negative elements on both secondary diagonals . Since they can be specified analytically, they can be used to test numerical inversion programs.

A one-to-one assignment between a permutation and a positive whole number in the faculty-based number representation ( factorial number system ) is referred to as the Lehmer code .

The Lehmer Five are those five natural numbers (276, 552, 564, 660, 966) below 1000 for which the asymptotic behavior of their “ aliquot sequence ”, the sequence of the iterated sum of all real divisors, has not yet been clarified.

The search for Cunningham chains also goes back to Lehmer. These are chains of prime numbers in which neighboring terms satisfy (the motivation comes from Lucas-type primality tests, in which the test whether it is prime has to be factored). Lehmer found three such chains of seven prime numbers in which the smallest prime number is less than . ${\displaystyle p_{i}=2\cdot p_{i-1}+1}$ ${\displaystyle p}$ ${\displaystyle (p-1)}$ ${\displaystyle 10^{7}}$

The close collaboration between Emma and Derrick Lehmer in number theory actually only has a counterpart in the case of the married couples Pierre and Marie Curie and their daughter Irène with her husband Frédéric Joliot-Curie in physics and chemistry.

He was the doctoral supervisor of Tom Apostol , John Brillhart , Ronald Graham , David Singmaster , Harold Stark and Peter Weinberger , among others .

Fonts

• Guide to Tables in the Theory of Numbers Washington, DC 1941, reprinted 1961
• Selected papers 1981.
• with John Brillhart , John L. Selfridge , Bryant Tuckerman , SS Wagstaff Jr .: Factorizations of ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers ${\displaystyle b^{n}}$ , American Mathematical Society 1983, 3 2nd edition, 2002
• An extended theory of Lucas' functions . Annals of Mathematics, 31, 419-448 (1930)
• A machine for combining sets of linear congruences . Mathematische Annalen, Vol. 109 (1934), pp. 661-667
• The Vanishing of Ramanujan's Function tau (n) . Duke Math. J., 14: 429-433 (1947)
• Mechanized mathematics . Bulletin of the American Mathematical Society, 1966.

literature

• Leo Corry : Number crunching vs. number theory: computers and FLT, from Kummer to SWAC (1850-1960), and beyond , Archive History Exact Sciences, 62, 2008, 393-455

Web links

• John J. O'Connor, Edmund F. Robertson :  Derrick Henry Lehmer. In: MacTutor History of Mathematics archive .
• Biography of John Brillhart in Acta Arithmetica (PDF; 191 kB)
• Leo Corry: Fermat Meets SWAC: Vandiver, the Lehmers, Computers, and Number Theory . IEEE Annals of the History of Computing 30 No. 1, 2008, pp. 38–49 (PDF; 442 kB)
• Lehmer exhibition at the University of Berkeley
• Lehmer at the University of Berkeley

Individual evidence

1. ^ RA Mollin: Number Theory and Applications. Banff 1988 Kluwer, Dordrecht, 1989, p. 194
2. ↑ See Ribenboim : The world of prime numbers , 1996, p. 80.
3. ↑ http://www.math.kent.edu/~varga/pub/paper_209.pdf
4. ^ DN Lehmer: Hunting big game in the theory of numbers. Scripta Mathematica 1, 1933, pp. 229–235 Entertaining description of the factorization as big game hunt
   also online as a copy by Richard Schroeppel
5. ↑ http://ed-thelen.org/comp-hist/Mike-Williams-Lehmer.html
6. ↑ Archived copy ( memento of the original dated February 23, 2010 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
7. ^ DH Lehmer: Euclid's algorithm for large numbers. Amer. Math. Monthly 45 (1938), pp. 227-233
8. ^ E. Bach, J. Shallit : Algorithmic Number Theory, Vol. I , MIT Press 1996, p. 300
9. ↑ Eric Weisstein: Lehmer's Constant . In: MathWorld (English).
10. ^ DH Lehmer: A machine method for solving polynomial equations. Journal ACM 8 (1961), pp. 151-162
11. ↑ I. Schur: About power series that are bounded inside the unit circle. Journal Reine Angew. Math. 148 (1918), pp. 122-145, especially p. 134
12. ↑ http://www.cecm.sfu.ca/~mjm/Lehmer/
13. ↑ Eriko Hironaka: What is Lehmer's number? (PDF file; 61 kB)
14. ↑ http://math.dartmouth.edu/~carlp/Carmichael_giuga.pdf
15. ↑ Richard GE Pinch; A note on Lehmer's totient problem Poster (PDF file; 48 kB) at ANTS VII (2006)
16. ^ DH Lehmer: Matrix paraphrases. Linear and Multilinear Algebra 28 (1991), No. 4, pp. 251-264
17. ↑ R. Mantaci, F. Rakotondrajao: A permutation representation that knows what "Eulerian" Means. (with reference to the original source; PDF; 92 kB)
18. ↑ http://www.aliquot.de/lehmer.htm
19. ↑ See Richard K. Guy Unsolved Problems in Number Theory , Springer Verlag 1994, p. 18
20. ↑ http://openlibrary.org/b/OL16601914M/Selected-papers-of-DH-Lehmer
21. ↑ http://www.ams.org/online_bks/conm22
22. ↑ http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002276976
23. ↑ http://www.ams.org/bull/1966-72-05/S0002-9904-1966-11553-7/home.html