# Questions on Pell's & Non-Linear Diophantine Equations UG Level

• MHB
• matqkks
In summary, Pell's equation is a type of Diophantine equation that involves finding integer solutions for variables. It specifically takes the form x^2 - Dy^2 = 1, where D is a non-square positive integer. A Non-Linear Diophantine equation is a type of Diophantine equation where the variables appear in non-linear terms, and they can have more complicated solutions than linear Diophantine equations. Pell's equation can be solved using various methods, and it has applications in number theory, cryptography, and physics. There are still many open problems and unsolved equations related to Pell's and Non-Linear Diophantine equations, with ongoing research in this area.
matqkks
Are there any resources of questions on the topic of Pell's and non linear Diophantine equations? I am looking for interesting results which are required to be solved. This is for a typical second year undergraduate level.
Thanks.

Have you looked at https://mathhelpboards.com/showthread.php?2905-The-Pell-Sequence in this forum?

## 1. What is Pell's equation?

Pell's equation is a type of Diophantine equation, which is an equation in which the variables are required to be integers. Pell's equation specifically takes the form of x^2 - Dy^2 = 1, where D is a non-square positive integer. It is named after the mathematician John Pell, who studied the equation in the 17th century.

## 2. What is a non-linear Diophantine equation?

A non-linear Diophantine equation is a type of Diophantine equation in which the variables are raised to a power other than 1. This means that the equations are not linear and cannot be solved using traditional algebraic methods. Non-linear Diophantine equations often have infinite solutions, making them challenging to solve.

## 3. What is the difference between Pell's equation and a non-linear Diophantine equation?

Pell's equation is a specific type of non-linear Diophantine equation. While Pell's equation takes the form of x^2 - Dy^2 = 1, non-linear Diophantine equations can take various forms, such as x^3 + y^3 = z^3 or x^2 + y^2 = z^5. Pell's equation is also unique in that it has a finite number of solutions, while non-linear Diophantine equations often have infinite solutions.

## 4. How are Pell's and non-linear Diophantine equations used in mathematics?

Pell's and non-linear Diophantine equations have various applications in mathematics, including number theory, algebraic geometry, and cryptography. They are also used in the study of prime numbers and in solving problems related to Fermat's Last Theorem.

## 5. What is the significance of studying Pell's and non-linear Diophantine equations at the undergraduate level?

Studying Pell's and non-linear Diophantine equations at the undergraduate level allows students to develop problem-solving skills and gain a deeper understanding of number theory and algebra. These equations also have practical applications in fields such as cryptography, making them relevant and valuable for students pursuing a career in mathematics or related fields.

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