Indeterminate system

<h2 id="examples">Examples</h2>
<h3>Linear indeterminate equations</h3>
An example linear indeterminate equation arises from imaging 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:

5
        y
        +
        8
        c
        +
        7
        n
        +
        90
        =
        7
        y
        +
        9
        c
        +
        6
        n
        +
        62
      
    
    {\displaystyle 5y+8c+7n+90=7y+9c+6n+62}

<a href="/facts/Bh%25C4%2581skara_II/BQlUY9gA">Bhāskara II</a> 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. 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
, resulting a series of possible solutions like (y, c, n)=(14, 1, 1), (13, 3, 1).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>: 43 
For given integers a, b and n, the general linear indeterminant equation is

a
        x
        +
        b
        y
        =
        n
      
    
    {\displaystyle ax+by=n}

with unknowns x and y restricted to integers. The necessary and sufficient condition for solutions is that the <a href="/facts/Greatest_common_divisor/kNFvBuKl">greatest common divisor</a>, 
 
 
 
 (
 a
 ,
 b
 )
 
 
 {\displaystyle (a,b)}
 
, is divisable by n.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>: 11 

<h2 id="history">History</h2>
Early mathematicians in both India and China studied indeterminate linear equations with integer solutions.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> Indian astronomer <a href="/facts/Aryabhata/3L1j5TDm">Aryabhata</a> developed a recursive algorithm to solve indeterminate equations now known to be related to <a href="/facts/Euclid%2527s_algorithm/wApql29c">Euclid's algorithm</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The name of the <a href="/facts/Chinese_remainder_theorem/7wwFIkuV">Chinese remainder theorem</a> 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.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
The first major work on indeterminate equations appears in <a href="/facts/Diophantus/XTWRgRWP">Diophantus</a>’ <a href="/facts/Arithmetica/wlPlkHvb">Arithmetica</a> in the 3rd century AD. Diophantus sought solutions constrained to be <a href="/facts/Rational_numbers/VJQmyY05">rational numbers</a>, but <a href="/facts/Pierre_de_Fermat/FLbUh75u">Pierre de Fermat</a>'s work in the 1600s focused on integer solutions and introduced the idea of characterizing all possible solutions rather than any one solution.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> In modern times integer solutions to indeterminate equations have come to be called analysis of <a href="/facts/Diophantine_equations/mYaHp7oX">Diophantine equations</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>: iii 
The original paper <a href="/facts/Henry_John_Stephen_Smith/sD12Jf6L">Henry John Stephen Smith</a> that defined the <a href="/facts/Smith_normal_form/pchMkW7Q">Smith normal form</a> was written for linear indeterminate systems.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hua, Luogeng (1982). "Chapter 11. Indeterminate Equations". Introduction to Number Theory. SpringerLink Bücher. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-642-68130-1. <a href="978-3-642-68130-1" target="_blank">978-3-642-68130-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Bashmakova, I. G. (1997). Diophantus and diophantine equations. Dolciani Mathematical Expositions. Washington, DC: Mathematical Association of America. ISBN 978-1-4704-5048-9. <a href="978-1-4704-5048-9" target="_blank">978-1-4704-5048-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dickson, L.E. (1919). History of the Theory of Numbers, Volume II: Diophantine Analysis. UK: Dover Publications (published 2013). <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dickson, L.E. (1919). History of the Theory of Numbers, Volume II: Diophantine Analysis. UK: Dover Publications (published 2013). <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Hua, Luogeng (1982). "Chapter 11. Indeterminate Equations". Introduction to Number Theory. SpringerLink Bücher. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-642-68130-1. <a href="978-3-642-68130-1" target="_blank">978-3-642-68130-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Christianidis, J. (1994). On the History of Indeterminate problems of the first degree in Greek Mathematics. Trends in the Historiography of Science, 237-247. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Shukla, K. N. (2015). The linear indeterminate equation-a brief historical account. Revista Brasileira de História da Matemática, 15(30), 83-94. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Christianidis, J. (1994). On the History of Indeterminate problems of the first degree in Greek Mathematics. Trends in the Historiography of Science, 237-247. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">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. <a href="https://www.tandfonline.com/doi/abs/10.1080/0025570X.2005.11953295" target="_blank">https://www.tandfonline.com/doi/abs/10.1080/0025570X.2005.11953295</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Dickson, L.E. (1919). History of the Theory of Numbers, Volume II: Diophantine Analysis. UK: Dover Publications (published 2013). <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Lazebnik, F. (1996). On systems of linear diophantine equations. Mathematics Magazine, 69(4), 261-266. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">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 href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Indeterminate system open-in-new

Indeterminate system