Finding minimal natural number solutions to linear Diophantine equations, using explicit quantification.
For a homogeneous problem, the solution is any linear combination of the resulting vectors. For a non-homogeneous problem, the solution is any linear combination of the vectors in the second component plus one of the vectors in the first component.
Solving diophantine equations
ldn: Solve a (L)inear (D)iophantine equation, returning minimal solutions over (N)aturals. The input is given as a rows of equations, with rhs values separated into a tuple.
Find the basis solution. By definition, the basis has all non-trivial (i.e., non-0) solutions that cannot be written as the sum of two other solutions. We use the mathematically equivalent statement that a solution is in the basis if it's least according to the lexicographic order using the ordinary less-than relation.
Solve the equation:
2x + y - z = 2
testNonHomogeneous [[0,2,0],[1,0,0]] [[0,1,1],[1,0,2]]
which means that the solutions are of the form:
(0, 2, 0) + k (0, 1, 1) + k' (1, 0, 2) = (k', 2+k, k+2k')
(1, 0, 0) + k (0, 1, 1) + k' (1, 0, 2) = (1+k', k, k+2k')
k'. It's easy to see that these are really solutions
to the equation given. It's harder to see that they cover all possibilities,
but a moments thought reveals that is indeed the case.
A puzzle: Five sailors and a monkey escape from a naufrage and reach an island with coconuts. Before dawn, they gather a few of them and decide to sleep first and share the next day. At night, however, one of them awakes, counts the nuts, makes five parts, gives the remaining nut to the monkey, saves his share away, and sleeps. All other sailors do the same, one by one. When they all wake up in the morning, they again make 5 shares, and give the last remaining nut to the monkey. How many nuts were there at the beginning?
We can model this as a series of diophantine equations:
x_0 = 5 x_1 + 1 4 x_1 = 5 x_2 + 1 4 x_2 = 5 x_3 + 1 4 x_3 = 5 x_4 + 1 4 x_4 = 5 x_5 + 1 4 x_5 = 5 x_6 + 1
We need to find to solve for x_0, over the naturals. We have:
* There was a total of 15621 coconuts * 1st sailor: 15621 = 3124*5+1, leaving 15621-3124-1 = 12496 * 2nd sailor: 12496 = 2499*5+1, leaving 12496-2499-1 = 9996 * 3rd sailor: 9996 = 1999*5+1, leaving 9996-1999-1 = 7996 * 4th sailor: 7996 = 1599*5+1, leaving 7996-1599-1 = 6396 * 5th sailor: 6396 = 1279*5+1, leaving 6396-1279-1 = 5116 * In the morning, they had: 5116 = 1023*5+1.
Note that this is the minimum solution, that is, we are guaranteed that there's
no solution with less number of coconuts. In fact, any member of
[15625*k-4 | k <- [1..]]
is a solution, i.e., so are