Count the number of integer solutions

In summary, the problem is to count the number of integer solutions for the inequality n+\sum_{k=1}^{n} \left| x_{k}\right| \leq N, where N' > 0. This can be solved by finding the number of lattice point solutions for \sum _{k=1} ^n |x_k| \leq N' and then using the recursive formula f(N',n) = f(N',n-1) + 2\sum _{k=1} ^{N'} f(N'-k,n-1).
  • #1
benorin
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Count the number of integer solutions of (rather, # of integer lattice points such that)

[tex]n+\sum_{k=1}^{n} \left| x_{k}\right| \leq N[/tex]

Not homework, so no rush. I have worked it through before with a prof., but he's so brilliant I didn't understand much of anything he said :redface: . His solution involed a generating function, something like the coefficient of the blah term in the expansion of blahblah.

Note that the LHS is the height of a polynomial with integer coefficients, although it has nothing to do with the solution.
 
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  • #2
If n > N, the answer is 0. Otherwise, the problem reduces to finding the lattice point solutions of:

[tex]\sum _{k=1} ^n |x_k| \leq N'[/tex]

where N' > 0. Let f(N',n) denote the number of solutions. Then we want to find f(N',n) and this is just:

[tex]f(N',n) = f(N',n-1) + 2\sum _{k=1} ^{N'} f(N'-k,n-1)[/tex]

We get this because [itex]\sum _{k=1} ^n |x_k| = |x_1| + \sum_{k=2} ^n |x_k|[/itex] and when |x1| > 0, then x1 has two possible values whereas when |x1| = 0 then x1 = 0 only. It's a start.
 
  • #3


To count the number of integer solutions for this equation, we need to find the number of integer lattice points that satisfy the inequality. This can be visualized as finding the number of lattice points that lie within or on the boundary of a specific region on the coordinate plane.

We can rewrite the equation as follows:
n + |x1| + |x2| + ... + |xn| ≤ N
This can be interpreted as finding all possible combinations of integers x1, x2, ..., xn such that their absolute values sum up to a value less than or equal to N-n.

To solve this problem, we can use generating functions, as mentioned by your professor. A generating function is a way to represent a sequence of numbers as coefficients in a power series. In this case, we can use the generating function (1 + x + x^2 + ...)^n to represent all possible combinations of integers x1, x2, ..., xn.

Expanding this generating function, we get (1 + x + x^2 + ...)^n = (1-x)^(-n).
Using the binomial theorem, we can write this as (1 + nx + n(n-1)x^2/2! + ...).
The coefficient of x^k in this expansion represents the number of ways to choose k terms from the sequence (1, 2, 3, ...) to get a sum of n.

In our case, we are interested in the coefficient of x^0, since that represents the number of ways to choose 0 terms from the sequence, which is just 1.
Thus, the number of integer solutions for our original equation is 1, which means there is only one lattice point that satisfies the inequality.

In summary, the number of integer solutions for the given equation is 1, as represented by the coefficient of x^0 in the generating function (1 + x + x^2 + ...)^n. This solution may seem counterintuitive, but it is a result of using generating functions to solve the problem.
 

What does it mean to "count the number of integer solutions"?

Counting the number of integer solutions refers to determining the number of whole number solutions to a mathematical equation or problem.

What types of equations or problems require counting the number of integer solutions?

Common examples include diophantine equations, which are polynomial equations where the solutions must be integers, and combinatorial problems such as counting the number of ways to arrange objects or combinations of objects.

How is counting the number of integer solutions different from finding the solutions themselves?

Counting the number of integer solutions involves determining the total number of solutions, while finding the solutions themselves involves determining the specific values of the solutions.

What strategies can be used to count the number of integer solutions?

Some common strategies include using algebraic techniques, such as factoring and substitution, to simplify equations and make it easier to count solutions. Combinatorial techniques, such as using permutations and combinations, can also be useful in counting solutions to certain types of problems.

Why is counting the number of integer solutions important in scientific research?

Counting the number of integer solutions is important in many areas of scientific research, including mathematics, computer science, and physics. It allows researchers to analyze and understand complex systems and phenomena, and can also help in making predictions and solving real-world problems.

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