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How do I generate a set of n-tuples containing integral solutions to a linear equation provided certain constraints?


Let m,k,p be fixed positive integers. I want to create a table of k-tuples (x1,x2,…,xk) comprised of solutions in positive integers to the equation below:


x1+x2+⋯+xk=mwhere 1≤x1,x2,…,xk≤p


For example, if I set m=16,k=3, and p=6. How can I make mathematica generate the set (table) of integer triples (x1,x2,x3) such that x1+x2+x3=16

where 1≤x1,x2,x3≤6.


Also, how would I seperately generate a table of positive integral triples (x1,x2,x3) for which x1+x2+x3≥16and 1≤x1,x2,x3≤6.



Answer



Try to read Documentation on functions Solve and Reduce and tutorial Solving Equations. Look through this forum, I bet this is a duplicate question. There is also a guide: Diophantine Equations.


{x1, x2, x3} /. 
 Solve[x1 + x2 + x3 == 16 && 1 <= x1 <= 6 && 1 <= x2 <= 6 && 

   1 <= x3 <= 6, {x1, x2, x3}, Integers]


{{4, 6, 6}, {5, 5, 6}, {5, 6, 5}, {6, 4, 6}, {6, 5, 5}, {6, 6, 4}}



or alternatively


Reduce[x1 + x2 + x3 == 16 && 1 <= x1 <= 6 && 1 <= x2 <= 6 && 
  1 <= x3 <= 6, {x1, x2, x3}, Integers]



(x1 == 4 && x2 == 6 && x3 == 6) || (x1 == 5 && x2 == 5 && x3 == 6) || (x1 == 5 && x2 == 6 && x3 == 5) || (x1 == 6 && x2 == 4 && x3 == 6) || (x1 == 6 && x2 == 5 && x3 == 5) || (x1 == 6 && x2 == 6 && x3 == 4)



Your 2nd question would be


{x1, x2, x3} /. 
 Solve[x1 + x2 + x3 >= 16 && 1 <= x1 <= 6 && 1 <= x2 <= 6 && 
   1 <= x3 <= 6, {x1, x2, x3}, Integers]


{{4, 6, 6}, {5, 5, 6}, {5, 6, 5}, {5, 6, 6}, {6, 4, 6}, {6, 5, 5}, {6, 5, 6}, {6, 6, 4}, {6, 6, 5}, {6, 6, 6}}




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