{{For|unconstrained systems of equations|Underdetermined system}}
In [[mathematics]], particularly in [[number theory]], an '''indeterminate system''' has fewer equations than unknowns but an additional set of constraints on the unknowns, such as restrictions that the values be integers.<ref name=Hua-1982>{{Cite book |last=Hua |first=Luogeng |title=Introduction to Number Theory |date=1982 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-68130-1 |series=SpringerLink Bücher |location=Berlin, Heidelberg |chapter=Chapter 11. Indeterminate Equations}}</ref> In modern times indeterminate equations are often called [[Diophantine equations]].<ref>{{Cite book |last=Bashmakova |first=I. G. |author-link=Isabella Bashmakova |title=Diophantus and diophantine equations |date=1997 |publisher=Mathematical Association of America |isbn=978-1-4704-5048-9 |series=Dolciani Mathematical Expositions |location=Washington, DC}}</ref><ref name="Dickson-1919">{{Cite book |last=Dickson |first=L.E. |title=History of the Theory of Numbers, Volume II: Diophantine Analysis. |date=1919 |publisher=Dover Publications |publication-date=2013 |location=UK}}</ref>{{rp|iii|q=Diophantine analysis was named after the Greek Diophantus, of the third century, who proposed many indeterminate problems in his arithmetic.}}

==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:
<math display="block">5y + 8c + 7n + 90 = 7y + 9c + 6n + 62</math>
[[Bhāskara II]] gave a general approach to this kind of problem by assigning a fixed integer to one (or N-2 in general) of the unknowns, e.g. <math>n=1</math>, resulting a series of possible solutions like (y, c, n)=(14, 1, 1), (13, 3, 1).<ref name=Dickson-1919/>{{rp|43}}

For given integers {{mvar|a}}, {{mvar|b}} and {{mvar|n}}, the general linear indeterminant equation is
<math display="block">ax + by = n</math>
with unknowns {{mvar|x}} and {{mvar|y}} restricted to integers. The necessary and sufficient condition for solutions is that the [[greatest common divisor]], <math>(a,b)</math>, is divisible by {{mvar|n}}.<ref name=Hua-1982/>{{rp|11}}

== History ==
Early mathematicians in both India and China studied indeterminate linear equations with integer solutions.<ref name=Christianidis>Christianidis, J. (1994). On the History of Indeterminate problems of the first degree in Greek Mathematics. Trends in the Historiography of Science, 237-247.</ref> Indian astronomer [[Aryabhata]] developed a recursive algorithm to solve indeterminate equations now known to be related to [[Euclid's algorithm]].<ref>Shukla, K. N. (2015). The linear indeterminate equation-a brief historical account. Revista Brasileira de História da Matemática, 15(30), 83-94.</ref> 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.<ref name=Christianidis/>

The first major work on indeterminate equations appears in [[Diophantus]]’ [[Arithmetica]] in the 3rd century AD. 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.<ref>{{Cite journal |last=Kleiner |first=Israel |date=2005-02-01 |title=Fermat: The Founder of Modern Number Theory |url=https://www.tandfonline.com/doi/abs/10.1080/0025570X.2005.11953295 |journal=Mathematics Magazine |volume=78 |issue=1 |pages=3–14 |doi=10.1080/0025570X.2005.11953295 |issn=0025-570X|url-access=subscription }}</ref> In modern times integer solutions to indeterminate equations have come to be called analysis of [[Diophantine equations]].<ref name=Dickson-1919/>{{rp|iii|q=Many later writers required solutions in integers (whole numbers), so that the term Diophantine analysis is used also in this altered sense.}}

The original paper [[Henry John Stephen Smith]] that defined the [[Smith normal form]] was written for linear indeterminate systems.<ref>Lazebnik, F. (1996). On systems of linear diophantine equations. Mathematics Magazine, 69(4), 261-266.</ref><ref>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.</ref>

==References==
{{reflist}}

[[Category:Number theory]]