In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers.
In modern times indeterminate equations are often called
Diophantine equations ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
.
Examples
Linear indeterminate equations
An example linear indeterminate equation arises from imagining two equally rich men, one with 5 rubies, 8 sapphires, 7 pearls and 90 gold coins; the other has 7, 9, 6 and 62 gold coins; find the prices (y, c, n) of the respective gems in gold coins. As they are equally rich:
Bhāskara II
Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferre ...
gave an general approach to this kind of problem by assigning a fixed integer to one (or N-2 in general) of the unknowns, e.g.
, resulting a series of possible solutions like (y, c, n)=(14, 1, 1), (13, 3, 1).
[
For given integers , and , the general linear indeterminant equation is
with unknowns and restricted to integers. The necessary and sufficient condition for solutions is that the ]greatest common divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...
, , is divisible by .[
]
History
Early mathematicians in both India and China studied indeterminate linear equations with integer solutions.[Christianidis, J. (1994). On the History of Indeterminate problems of the first degree in Greek Mathematics. Trends in the Historiography of Science, 237-247.] Indian astronomer Aryabhata
Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...
developed a recursive algorithm to solve indeterminate equations now known to be related to Euclid's algorithm. The name of the Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
relates to the view that indeterminate equations arose in these asian mathematical traditions, but it is likely that ancient Greeks also worked with indeterminate equations.[
The first major work on indeterminate equations appears in ]Diophantus
Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations.
Although Jose ...
’ Arithmetica
Diophantus of Alexandria () (; ) was a Greek mathematics, Greek mathematician who was the author of the ''Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations ...
in the 3rd century AD. Diophantus sought solutions constrained to be rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
, but Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
's work in the 1600s focused on integer solutions and introduced the idea of characterizing all possible solutions rather than any one solution. In modern times integer solutions to indeterminate equations have come to be called analysis of Diophantine equations ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
.[
The original paper Henry John Stephen Smith that defined the Smith normal form was written for linear indeterminate systems.][Smith, H. J. S. (1861). Xv. on systems of linear indeterminate equations and congruences. Philosophical transactions of the royal society of london, (151), 293-326.]
References
{{reflist
Number theory