Results 131 to 140 of about 661 (172)
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2014
In February, 1657, Fermat challenged the English mathematicians John Wallis (1616–1703) and Lord William V. Brouncker (1620–1684) to solve the non-linear diophantine equation \(x^{2} - dy^{2} = 1\), where d is nonsquare and positive. The amateur French mathematician Bernard de Bessey (ca. 1605–1675) solved it for d ≤ 150.
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In February, 1657, Fermat challenged the English mathematicians John Wallis (1616–1703) and Lord William V. Brouncker (1620–1684) to solve the non-linear diophantine equation \(x^{2} - dy^{2} = 1\), where d is nonsquare and positive. The amateur French mathematician Bernard de Bessey (ca. 1605–1675) solved it for d ≤ 150.
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2021
We give the classical proof of the solvability of the Pell equation, present a method for computing the fundamental unit, and show how to apply these results to the determination of squares in Lucas sequences.
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We give the classical proof of the solvability of the Pell equation, present a method for computing the fundamental unit, and show how to apply these results to the determination of squares in Lucas sequences.
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PELL’S EQUATIONS IN GAUSSIAN INTEGERS
JP Journal of Algebra, Number Theory and Applications, 2019Summary: From Hurwitz's approach of complex continued fractions, we build a complex theory of the Pell's equation. In this paper, we study the complex theory of the Pell's equation, \(x^2-Dy^2=2\), that is, finding its solutions in Gaussian integers, using Hurwitz complex continued fraction, hence, generalizing it to the Pell's equation \(x^2-Dy^2=2^n\)
Kharbuki, Algracia, Singh, Madan Mohan
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2003
The so-called Pell equation x2 − ny2 = 1 (wrongly attributed to Pell by Euler) is one of the oldest equations in mathematics and it is fundamental to the study of quadratic Diophantine equations. The Greeks studied the special case x2 − 2y2 = 1 because they realized that its natural number solutions throw light on the nature of \(\sqrt{2}\). There is a
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The so-called Pell equation x2 − ny2 = 1 (wrongly attributed to Pell by Euler) is one of the oldest equations in mathematics and it is fundamental to the study of quadratic Diophantine equations. The Greeks studied the special case x2 − 2y2 = 1 because they realized that its natural number solutions throw light on the nature of \(\sqrt{2}\). There is a
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2023
The purpose of this paper is to develop a method for finding values of N for which the equation x[squared] - DY[squared] = N will have integral solution in terms of X and Y for a given value of D which is a natural number, not a perfect square. It is shown that values of N are determined by the particular form of D.
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The purpose of this paper is to develop a method for finding values of N for which the equation x[squared] - DY[squared] = N will have integral solution in terms of X and Y for a given value of D which is a natural number, not a perfect square. It is shown that values of N are determined by the particular form of D.
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2002
Summary: Since the time of Archimedes, mathematicians have challenged their peers with a Diophantine equation that is satisfied by huge integers. The article discusses the algorithmic efficiency of various solution methods.
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Summary: Since the time of Archimedes, mathematicians have challenged their peers with a Diophantine equation that is satisfied by huge integers. The article discusses the algorithmic efficiency of various solution methods.
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The Set of Automorphisms of Pell Forms and Pell Equations
2022d = k(2) + 4 for some integer k >= 2. In this work, we first determined the set of automorphisms of the Pell form F-delta(x, y) = x(2) - dy(2) of discriminant delta = 4d. Later, we deduced the set of all integer solutions of the Pell equations F-delta(x, y) = +/- 1 and F-delta(x, y) = +/- k(2).
TEKCAN, AHMET, Biberoglu, Gulsah
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2009
In 1909, A. Thue proved the following important theorem: Let f = a n z n +a n -1zn-1+…+a1z+a0 be an irreducible polynomial of degree ≥ 3 with integral coefficients. Consider the corresponding homogeneous polynomial $$F(x,y)=y^nf\left(\frac{x}{y}\right)=a_nx^n+a_{n-1}x^{n-1}y+\cdots+a_1xy^{n-1}+a_0y^n.$$
Titu Andreescu +2 more
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In 1909, A. Thue proved the following important theorem: Let f = a n z n +a n -1zn-1+…+a1z+a0 be an irreducible polynomial of degree ≥ 3 with integral coefficients. Consider the corresponding homogeneous polynomial $$F(x,y)=y^nf\left(\frac{x}{y}\right)=a_nx^n+a_{n-1}x^{n-1}y+\cdots+a_1xy^{n-1}+a_0y^n.$$
Titu Andreescu +2 more
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Journal of Mathematical Sciences, 2011
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