Daniel Shanks
Encyclopedia
Daniel Shanks was an American
mathematician
who worked primarily in numerical analysis
and number theory
. He is best known as the first to compute π
to 100,000 decimal places, and for his book Solved and Unsolved Problems in Number Theory.
, who was also known for computation of π. He earned his Bachelor of Science
degree in physics from the University of Chicago
in 1937 and a Ph.D.
in Mathematics from the University of Maryland
in 1954. In between these two, Shanks worked at the Aberdeen Proving Ground
and the Naval Ordnance Laboratory
, first as a physicist and then as a mathematician. During this period he also wrote his Ph.D. thesis (completed in 1949), despite having never taken any graduate math courses.
After earning his Ph.D. in mathematics, Shanks continued working at the Naval Ordnance Laboratory
and the Naval Ship Research and Development Center at the David Taylor Model Basin
, where he stayed until 1976. He then spent a year at National Bureau of Standards before moving to the University of Maryland
as an adjunct professor. He remained in Maryland for the rest of his life.
Dan Shanks died on September 6, 1996.
and number theory
, but he had many interests and also did some work in black body
radiation, ballistics
, mathematical identities, and Epstein zeta functions.
and others to compute the number π
to 100,000 decimals on a computer.
This was done in 1961, and it was a major advance over previous work.
Shanks was an editor of the Mathematics of Computation
from 1959 until his death. He was noted for his very thorough reviews of papers, and for being a jack-of-all-trades who did whatever was necessary to get the journal out.
Hugh Williams described it as "a charming, unconventional, provocative, and fascinating book on elementary number theory." It is a wide-ranging book, but most of the topics depend on quadratic residues and Pell's equation
. The third edition contains a long essay on "judging conjectures." Shanks contended that there should be a lot of evidence that something is true before we classify it as a conjecture (otherwise it should be an Open Question and we should not take sides on it), and his essay gives many examples of bad thinking deriving from premature conjecturing. Writing about the possible non-existence of odd perfect numbers, 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
. He developed a number of fast computer factorization methods based on quadratic forms and the class number. His algorithms include: Baby-step giant-step
algorithm for computing the discrete logarithm
, which is useful in public-key cryptography
; Shanks' square forms factorization
, an integer factorization
method that generalizes Fermat's factorization method
; and the Tonelli–Shanks algorithm that finds square roots moduli a prime, which is useful for the quadratic sieve
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, the sum of the reciprocals of the twin prime
s, calculating it over the twin primes among the first two million primes.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who worked primarily in numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. He is best known as the first to compute π
Numerical approximations of π
This page is about the history of approximations for the mathematical constant pi . There is a table summarizing the chronology of computation of π. See also the history of pi for other aspects of the evolution of our knowledge about mathematical properties of pi...
to 100,000 decimal places, and for his book Solved and Unsolved Problems in Number Theory.
Life and education
Dan (he insisted that everyone call him Dan) Shanks was born on January 17, 1917, in Chicago, Illinois, and he is not related to the English mathematician William ShanksWilliam Shanks
William Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F...
, who was also known for computation of π. He earned his Bachelor of Science
Bachelor of Science
A Bachelor of Science is an undergraduate academic degree awarded for completed courses that generally last three to five years .-Australia:In Australia, the BSc is a 3 year degree, offered from 1st year on...
degree in physics from the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
in 1937 and a Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
in Mathematics from the University of Maryland
University of Maryland, College Park
The University of Maryland, College Park is a top-ranked public research university located in the city of College Park in Prince George's County, Maryland, just outside Washington, D.C...
in 1954. In between these two, Shanks worked at the Aberdeen Proving Ground
Aberdeen Proving Ground
Aberdeen Proving Ground is a United States Army facility located near Aberdeen, Maryland, . Part of the facility is a census-designated place , which had a population of 3,116 at the 2000 census.- History :...
and the Naval Ordnance Laboratory
Naval Ordnance Laboratory
The Naval Ordnance Laboratory , now disestablished, formerly located in White Oak, Maryland was the site of considerable work that had practical impact upon world technology. The White Oak site of NOL has now been taken over by the Food and Drug Administration.-History:The U.S...
, first as a physicist and then as a mathematician. During this period he also wrote his Ph.D. thesis (completed in 1949), despite having never taken any graduate math courses.
After earning his Ph.D. in mathematics, Shanks continued working at the Naval Ordnance Laboratory
Naval Ordnance Laboratory
The Naval Ordnance Laboratory , now disestablished, formerly located in White Oak, Maryland was the site of considerable work that had practical impact upon world technology. The White Oak site of NOL has now been taken over by the Food and Drug Administration.-History:The U.S...
and the Naval Ship Research and Development Center at the David Taylor Model Basin
David Taylor Model Basin
The David Taylor Model Basin is one of the largest ship model basins — test facilities for the development of ship design — in the world...
, where he stayed until 1976. He then spent a year at National Bureau of Standards before moving to the University of Maryland
University of Maryland, College Park
The University of Maryland, College Park is a top-ranked public research university located in the city of College Park in Prince George's County, Maryland, just outside Washington, D.C...
as an adjunct professor. He remained in Maryland for the rest of his life.
Dan Shanks died on September 6, 1996.
Works
Shanks worked primarily in numerical analysisNumerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, but he had many interests and also did some work in black body
Black body
A black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...
radiation, ballistics
Ballistics
Ballistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...
, mathematical identities, and Epstein zeta functions.
Numerical analysis
Shanks's most prominent work in numerical analysis was a collaboration with John WrenchJohn Wrench
John William Wrench, Jr. was an American mathematician who worked primarily in numerical analysis. He was a pioneer in using computers for mathematical calculations, and is noted for work done with Daniel Shanks to calculate the mathematical constant pi to 100,000 decimal places.-Life and...
and others to compute the number π
Numerical approximations of π
This page is about the history of approximations for the mathematical constant pi . There is a table summarizing the chronology of computation of π. See also the history of pi for other aspects of the evolution of our knowledge about mathematical properties of pi...
to 100,000 decimals on a computer.
This was done in 1961, and it was a major advance over previous work.
Shanks was an editor of the Mathematics of Computation
Mathematics of Computation
Mathematics of Computation is a quarterly mathematics journal focused on computational mathematics that is published by the American Mathematical Society. It was established in 1943....
from 1959 until his death. He was noted for his very thorough reviews of papers, and for being a jack-of-all-trades who did whatever was necessary to get the journal out.
Number theory
In number theory, Shanks is best known for his book Solved and Unsolved Problems in Number Theory.Hugh Williams described it as "a charming, unconventional, provocative, and fascinating book on elementary number theory." It is a wide-ranging book, but most of the topics depend on quadratic residues and Pell's equation
Pell's equation
Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...
. The third edition contains a long essay on "judging conjectures." Shanks contended that there should be a lot of evidence that something is true before we classify it as a conjecture (otherwise it should be an Open Question and we should not take sides on it), and his essay gives many examples of bad thinking deriving from premature conjecturing. Writing about the possible non-existence of odd perfect numbers, 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
Computational number theory
In mathematics, computational number theory, also known as algorithmic number theory, is the study of algorithms for performing number theoretic computations...
. He developed a number of fast computer factorization methods based on quadratic forms and the class number. His algorithms include: Baby-step giant-step
Baby-step giant-step
In group theory, a branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm computing the discrete logarithm. The discrete log problem is of fundamental importance to the area of public key cryptography...
algorithm for computing the discrete logarithm
Discrete logarithm
In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. In particular, an ordinary logarithm loga is a solution of the equation ax = b over the real or complex numbers...
, which is useful in public-key cryptography
Public-key cryptography
Public-key cryptography refers to a cryptographic system requiring two separate keys, one to lock or encrypt the plaintext, and one to unlock or decrypt the cyphertext. Neither key will do both functions. One of these keys is published or public and the other is kept private...
; Shanks' square forms factorization
Shanks' square forms factorization
Shanks' square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method....
, an integer factorization
Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
method that generalizes Fermat's factorization method
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.\...
; and the Tonelli–Shanks algorithm that finds square roots moduli a prime, which is useful for the quadratic sieve
Quadratic sieve
The quadratic sieve algorithm is a modern integer factorization algorithm and, in practice, the second fastest method known . It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve...
method of integer factorization
Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
.
In 1974, Shanks and John Wrench
John Wrench
John William Wrench, Jr. was an American mathematician who worked primarily in numerical analysis. He was a pioneer in using computers for mathematical calculations, and is noted for work done with Daniel Shanks to calculate the mathematical constant pi to 100,000 decimal places.-Life and...
did some of the first computer work on estimating the value of Brun's constant, the sum of the reciprocals of the twin prime
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...
s, calculating it over the twin primes among the first two million primes.