Positive solution for linear Diophantine equations

In summary, the linear Diophantine equations in the form ax+by=c with natural numbers a, b, and c may have infinite integer solutions if c is a multiple of the greatest common divisor of a and b. However, finding positive integer solutions for x and y may not always be possible. All solutions of the equation are of the form x = x0 + kb and y = y0 - ka, where k is any integer. To have positive solutions, k must be chosen carefully based on the values of a, b, x0, and y0.
  • #1
pyfgcr
22
0
The linear Diophantine equations: ax+by=c, a,b,c is natural numbers.
If c is a multiple of gcd(a,b), there is infinite integer solutions, and I know how to find x,y.
However, I wonder how to find positive integer solution x,y only.
 
Mathematics news on Phys.org
  • #2
pyfgcr said:
The linear Diophantine equations: ax+by=c, a,b,c is natural numbers.
If c is a multiple of gcd(a,b), there is infinite integer solutions, and I know how to find x,y.
However, I wonder how to find positive integer solution x,y only.


They may not exists. For example, the equation [itex]\,7x+6y=5\,[/itex] cannot have positive solutions, but it has

solutions, like [itex]\,(5,-6)\,[/itex]

DonAntonio
 
  • #3
Shouldn't that be (5,-5)?
 
  • #4
Mensanator said:
Shouldn't that be (5,-5)?


Yes, you're right of course. Thanks.

DonAntonio
 
  • #5
All solutions of the Diophantine equation ax+ by= c (assuming a, b, relatively prime) are of the form x= x0+ kb, y= y0- ka for k any integer. If you want both x and y positive, you must be able to choose k so that those are postive. Whether that is possible, of course, depends on a, b, x0, and y0.
 

What is a linear Diophantine equation?

A linear Diophantine equation is an equation in two or more variables where the coefficients and constants are all integers.

What is a positive solution for a linear Diophantine equation?

A positive solution for a linear Diophantine equation is a set of values for the variables that satisfy the equation and all variables are positive integers.

How do you find positive solutions for a linear Diophantine equation?

To find positive solutions for a linear Diophantine equation, you can use methods such as substitution, elimination, or the Euclidean algorithm. These methods involve manipulating the equation to isolate the variables and finding values that satisfy the equation.

Can all linear Diophantine equations have positive solutions?

No, not all linear Diophantine equations have positive solutions. Some equations may only have negative or zero solutions, while others may have no solutions at all.

What are some real-world applications of linear Diophantine equations?

Linear Diophantine equations have many real-world applications, such as in cryptography, engineering, and economics. They can be used to model and solve problems involving proportions, ratios, and constraints in various fields.

Similar threads

Replies
4
Views
668
  • General Math
Replies
3
Views
958
  • General Math
Replies
6
Views
1K
Replies
2
Views
999
Replies
2
Views
2K
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
Replies
2
Views
903
  • General Math
Replies
0
Views
620
  • General Math
Replies
11
Views
1K
Back
Top