Do You Know Pell's Equation?

1) Who made the first significant progress in solving the Pell's equation?

Mahavira

Varahamihira

Yativṛṣabha

Brahmagupta

Brahmagupta made the first significant progress in solving the Pell's equation.

Explanation

Brahmagupta made the first significant progress in solving the Pell's equation.

2) In Cartesian coordinates, what form does the Pell's equation have?

Ellipse

Circle

Hyperbola

Parabola

not-available-via-ai

Explanation

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3) Where was the equation studied extensively for the first time?

India

Italy

Germany

England

India is the correct answer because the question asks where the equation was studied extensively for the first time. India is known for its rich history in mathematics, with scholars like Aryabhata and Brahmagupta making significant contributions to the field. The concept of equations and their study would have been explored extensively in ancient India, making it the likely place where the equation was first studied in depth.

Explanation

India is the correct answer because the question asks where the equation was studied extensively for the first time. India is known for its rich history in mathematics, with scholars like Aryabhata and Brahmagupta making significant contributions to the field. The concept of equations and their study would have been explored extensively in ancient India, making it the likely place where the equation was first studied in depth.

4) What did Bhāskarāchārya developed for solving Pell's equation?

Chakravala method

Geometric method

Graphic method

Muller's method

Bhāskarāchārya developed the Chakravala method for solving Pell's equation. This method is a mathematical algorithm that involves manipulating and simplifying the equation using a series of steps. It is particularly useful for finding integer solutions to Pell's equation, which is a type of Diophantine equation. The Chakravala method has been widely recognized for its efficiency and effectiveness in solving this type of equation.

Explanation

Bhāskarāchārya developed the Chakravala method for solving Pell's equation. This method is a mathematical algorithm that involves manipulating and simplifying the equation using a series of steps. It is particularly useful for finding integer solutions to Pell's equation, which is a type of Diophantine equation. The Chakravala method has been widely recognized for its efficiency and effectiveness in solving this type of equation.

5) With which subject in mathematics does Pell's equation have connections?

Chebyshev polynomials

Continued fractions

Algebraic number theory

All of the above

Pell's equation is a mathematical equation that is connected to all three subjects mentioned: Chebyshev polynomials, continued fractions, and algebraic number theory. Chebyshev polynomials are used in solving Pell's equation, continued fractions can be used to approximate solutions to Pell's equation, and algebraic number theory provides a deeper understanding of the properties and solutions of Pell's equation. Therefore, the correct answer is "All of the above."

Explanation

Pell's equation is a mathematical equation that is connected to all three subjects mentioned: Chebyshev polynomials, continued fractions, and algebraic number theory. Chebyshev polynomials are used in solving Pell's equation, continued fractions can be used to approximate solutions to Pell's equation, and algebraic number theory provides a deeper understanding of the properties and solutions of Pell's equation. Therefore, the correct answer is "All of the above."

6) Who has mistakenly attributed Lord Brouncker's solution of the equation to John Pell?

Johann Bernoulli

Leonhard Euler

John Pell

Pierre Bouguer

Leonhard Euler mistakenly attributed Lord Brouncker's solution of the equation to John Pell.

Explanation

Leonhard Euler mistakenly attributed Lord Brouncker's solution of the equation to John Pell.

7) Who was the first European that found the solution for Pell's equation?

William Brouncker

John Pell

Bathsua Makin

Leonhard Euler

William Brouncker is the correct answer because he was the first European mathematician to find the solution for Pell's equation. Pell's equation is a mathematical equation of the form x^2 - ny^2 = 1, where n is a positive nonsquare integer. Brouncker's solution to this equation was published in 1657, making him the first to find a general solution for it. John Pell, Bathsua Makin, and Leonhard Euler were not specifically known for their contributions to Pell's equation.

Explanation

William Brouncker is the correct answer because he was the first European mathematician to find the solution for Pell's equation. Pell's equation is a mathematical equation of the form x^2 - ny^2 = 1, where n is a positive nonsquare integer. Brouncker's solution to this equation was published in 1657, making him the first to find a general solution for it. John Pell, Bathsua Makin, and Leonhard Euler were not specifically known for their contributions to Pell's equation.

8) What is the Brouncker's solution for the equation (see picture) that was made by Fermat in 1657 to chalenge his fellows mathematicians?

(x,y) = (32188120829134849,181938015856460)

(x,y) = (1,2)

(x,y) = (176619049,226153980)

(x,y) = (0,-1)

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Explanation

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9) When was William Brouncker born?

1596

1605

1620

1643

William Brouncker was born in 1620.

Explanation

William Brouncker was born in 1620.

10) Who was the first European that concluded that if n is not a perfect square, Pell's equation can have infinite distinct integer solutions?

John Pell

Narayana Pandit

William Brouncker

Joseph Louis Lagrange

Joseph Louis Lagrange was a renowned mathematician who made significant contributions to number theory, including the study of Pell's equation. Pell's equation is a Diophantine equation of the form x^2 - ny^2 = 1, where n is a non-square positive integer. Lagrange concluded that if n is not a perfect square, Pell's equation can have infinite distinct integer solutions. His work on Pell's equation laid the foundation for further research and understanding of the equation, making him the first European to reach this conclusion.

Explanation

Joseph Louis Lagrange was a renowned mathematician who made significant contributions to number theory, including the study of Pell's equation. Pell's equation is a Diophantine equation of the form x^2 - ny^2 = 1, where n is a non-square positive integer. Lagrange concluded that if n is not a perfect square, Pell's equation can have infinite distinct integer solutions. His work on Pell's equation laid the foundation for further research and understanding of the equation, making him the first European to reach this conclusion.