Indeterminate system
In mathematics, particularly in number theory, 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.[1]
Examples
Linear indeterminate equations
For given integers a, b and n, a linear indeterminant equation is with unknowns x and y restricted to integers. The necessary and sufficient condition for solutions is that the greatest common divisor, , is divisable by n.[1]: 11
Smith normal form
The original paper Henry John Stephen Smith that defined the Smith normal form was written for linear indeterminate systems.[2][3]
History
Early mathematicians in both India and China studied indeterminate linear equations with integer solutions.[4] Indian astronomer Aryabhata developed a recursive algorithm to solve indeterminate equations now known to be related to Euclid's algorithm.[5] The name of the Chinese remainder theorem 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.[4]
The first major work on indeterminate equations appears in Diophantus’ Arithmetica in the 3rd century AD. In modern times indeterminate equations have come to be called Diophantine equations. Diophantus sought solutions constrained to be rational numbers, but Pierre de Fermat's work in the 1600s focused on integer solutions and introduced the idea of characterizing all possible solutions rather than any one solution.[6]
References
- ^ a b Hua, Luogeng (1982). "Chapter 11. Indeterminate Equations". Introduction to Number Theory. SpringerLink Bücher. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-642-68130-1.
- ^ Lazebnik, F. (1996). On systems of linear diophantine equations. Mathematics Magazine, 69(4), 261-266.
- ^ 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.
- ^ a b Christianidis, J. (1994). On the History of Indeterminate problems of the first degree in Greek Mathematics. Trends in the Historiography of Science, 237-247.
- ^ Shukla, K. N. (2015). The linear indeterminate equation-a brief historical account. Revista Brasileira de História da Matemática, 15(30), 83-94.
- ^ Kleiner, Israel (2005-02-01). "Fermat: The Founder of Modern Number Theory". Mathematics Magazine. 78 (1): 3–14. doi:10.1080/0025570X.2005.11953295. ISSN 0025-570X.
Further reading
- Lay, David (2003). Linear Algebra and Its Applications. Addison-Wesley. ISBN 0-201-70970-8.