## Welcome to Mathematica Stack Exchange Mathematica® Stack Exchange is a question and answer site for users of Wolfram Mathematica and the Wolfram Language. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detailed answers to every question about Mathematica.

We're a little bit different from other sites. Here's how:

This site is all about getting answers. It's not a discussion forum. There's no chit-chat.

Just questions...

Good answers are voted up and rise to the top.

The best answers show up first so that they are always easy to find.

Accepting doesn't mean it's the best answer, it just means that it worked for the person who asked.

# Finding the number of solutions to a diophantine equation

I want to count total number of the natural solutions (different from 0) of the equation $2x + 3y + z = 100$, but don't know how. How can I calculate it using Mathematica? I tried:

Solve[{2*x + 3*y + z == 100, x > 0, y > 0, z > 0}, {x, y, z}, Integers]


There are also power series methods for counting these.

SeriesCoefficient[
x^(1 + 2 + 3)/(1 - x^1)*1/(1 - x^2)*1/(1 - x^3), {x, 0, 100}]

(* Out= 784 *)


This should work:

sols = Solve[{2*x + 3*y + z == 100, x > 0, y > 0, z > 0},
{x, y, z}, Integers];
Length@sols


## Get answers to practical, detailed questions

Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.

• Specific issues with Wolfram Mathematica
• Real problems or questions that you’ve encountered

Not all questions work well in our format. Avoid questions that are primarily opinion-based, or that are likely to generate discussion rather than answers.

Questions that need improvement may be closed until someone fixes them.

• Anything not directly related to Wolfram Mathematica
• Questions that are primarily opinion-based
• Questions with too many possible answers or that would require an extremely long answer

## Tags make it easy to find interesting questions

All questions are tagged with their subject areas. Each can have up to 5 tags, since a question might be related to several subjects.

Click any tag to see a list of questions with that tag, or go to the tag list to browse for topics that interest you.

# Finding the number of solutions to a diophantine equation

I want to count total number of the natural solutions (different from 0) of the equation $2x + 3y + z = 100$, but don't know how. How can I calculate it using Mathematica? I tried:

Solve[{2*x + 3*y + z == 100, x > 0, y > 0, z > 0}, {x, y, z}, Integers]


## You earn reputation when people vote on your posts

+10 question voted up
+2 edit approved

As you earn reputation, you'll unlock new privileges like the ability to vote, comment, and even edit other people's posts.

Reputation Privilege
15 Vote up
125 Vote down (costs 1 rep on answers)

At the highest levels, you'll have access to special moderation tools. You'll be able to work alongside our community moderators to keep the site focused and helpful.

Reputation Privilege
2000 Edit other people's posts
3000 Vote to close, reopen, or migrate questions
see all privileges

## Improve posts by editing or commenting

Our goal is to have the best answers to every question, so if you see questions or answers that can be improved, you can edit them.

Use edits to fix mistakes, improve formatting, or clarify the meaning of a post.

You can always comment on your own questions and answers. Once you earn 50 reputation, you can comment on anybody's post.

Remember: we're all here to learn, so be friendly and helpful!

There are also power series methods for counting these.

SeriesCoefficient[
x^(1 + 2 + 3)/(1 - x^1)*1/(1 - x^2)*1/(1 - x^3), {x, 0, 100}]

(* Out= 784 *)

Generating functions can be nice sometimes... - J. M.'s eventual burnout Apr 17, 2013 at 20:38

## Unlock badges for special achievements

Badges are special achievements you earn for participating on the site. They come in three levels: bronze, silver, and gold.

 Informed Read the entire tour page
 Student First question with score of 1 or more Editor First edit Good Answer Answer score of 25 or more Civic Duty Vote 300 or more times Famous Question Question with 10,000 views 