Daniel Shanks (January 17, 1917 – September 6, 1996) was an

American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, pe ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who worked primarily in

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

and

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

. He was the first person to compute π to 100,000 decimal places.

Life and education

Shanks was born on January 17, 1917, in

Chicago, Illinois (''City in a Garden''); I Will , image_map = , map_caption = Interactive Map of Chicago , coordinates = , coordinates_footnotes = , subdivision_type = Country , subdivision_name ...

. He is not related to the English mathematician

William Shanks William Shanks (25 January 1812 – June 1882) was an English amateur mathematician. He is famous for his calculation of '' '' (pi) to 707 places in 1873, which was correct up to the first 527 places. The error was discovered in 1944 by D. F. Fe ...

, who was also known for his computation of π. He earned his

Bachelor of Science A Bachelor of Science (BS, BSc, SB, or ScB; from the Latin ') is a bachelor's degree awarded for programs that generally last three to five years. The first university to admit a student to the degree of Bachelor of Science was the University o ...

degree in physics from the

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...

in 1937, and a

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

in Mathematics from the

University of Maryland The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public land-grant research university in College Park, Maryland. Founded in 1856, UMD is the flagship institution of the University System of M ...

in 1954. Prior to obtaining his PhD, Shanks worked at the

Aberdeen Proving Ground Aberdeen Proving Ground (APG) (sometimes erroneously called Aberdeen Proving ''Grounds'') is a U.S. Army facility located adjacent to Aberdeen, Harford County, Maryland, United States. More than 7,500 civilians and 5,000 military personnel work a ...

and the

Naval Ordnance Laboratory The Naval Ordnance Laboratory (NOL) was a facility in the White Oak area of Montgomery County, Maryland. It is now used as the headquarters of the U.S. Food and Drug Administration. Origins The U.S. Navy Mine Unit, later the Mine Laboratory at ...

, first as a physicist and then as a mathematician. During this period he wrote his PhD thesis, which completed in 1949, despite having never taken any graduate math courses. After earning his PhD in mathematics, Shanks continued working at the

and the Naval Ship Research and Development Center at

David Taylor Model Basin The David Taylor Model Basin (DTMB) is one of the largest ship model basins—test facilities for the development of ship design—in the world. DTMB is a field activity of the Carderock Division of the Naval Surface Warfare Center. Hist ...

, where he stayed until 1976. He spent one year at the

National Bureau of Standards The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...

before moving to the

as an adjunct professor. He remained in Maryland for the rest of his life. Shanks died on September 6, 1996.

Works

Shanks worked primarily in

and

; however, he had many interests and also worked on

black body A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...

radiation, ballistics, mathematical identities, and

Epstein zeta function In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are ma ...

Numerical analysis

Shanks's most prominent work in numerical analysis was a collaboration with John Wrench and others to compute the number π to 100,000 decimal digits on a computer. This was done in 1961 on an

IBM 7090 The IBM 7090 is a second-generation transistorized version of the earlier IBM 709 vacuum tube mainframe computer that was designed for "large-scale scientific and technological applications". The 7090 is the fourth member of the IBM 700/7000 se ...

, and it was a major advancement over previous work. Shanks was an editor of the ''

Mathematics of Computation ''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than fiv ...

'' from 1959 until his death. He was noted for his very thorough reviews of papers, and for doing whatever was necessary to get the journal out.

Number theory

Shanks wrote the book ''Solved and Unsolved Problems in Number Theory'', which mostly depended on

quadratic residues In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv q \pmod. Otherwise, ''q'' is called a quadratic non ...

and

Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...

. The third edition of the book contains a long essay on judging conjectures, in which Shanks contended that unless there is a lot of evidence to suggest that something is true, it should not be classified as a conjecture, but rather as an open question. His essay provided many examples of bad thinking that were derived from premature conjecturing. Writing about the possible non-existence of odd

perfect numbers In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...

, which had been checked to 10⁵⁰, he famously remarked that "10⁵⁰ is a long way from infinity." Most of Shanks's number theory work was in

computational number theory In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithm ...

. He developed a number of fast computer factorization methods based on

quadratic forms In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

and the class number. His

algorithms In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

include:

Baby-step giant-step In group theory, a branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm or order of an element in a finite abelian group by Daniel Shanks. The discrete log problem is of fundamenta ...

algorithm for computing the discrete logarithm, which is useful in

public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...

;

Shanks's square forms factorization Shanks's square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success of Fermat's method depends on finding integers x and y such that x^2-y^2=N, where ...

, integer factorization method that generalizes

Fermat's factorization method Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: :N = a^2 - b^2. That difference is algebraically factorable as (a+b)(a-b); if neither factor equals one, ...

; and the

Tonelli–Shanks algorithm The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for ''r'' in a congruence of the form ''r''2 ≡ ''n'' (mod ''p''), where ''p'' is a prime: that is, to find a square root of ''n'' ...

that finds square roots modulo a prime, which is useful for the

quadratic sieve The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is consider ...

method of integer factorization. In 1974, Shanks and John Wrench did some of the first computer work on estimating the value of

Brun's constant In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by V ...

, the sum of the reciprocals of the

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...

s, calculating it over the twin primes among the first two million primes.

Notes

External links