Results 161 to 170 of about 20,503 (203)
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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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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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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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2015
This chapter gives the general theory and useful algorithms to find positive integer solutions (x, y) to general Pell’s equation (4.1.1), where D is a nonsquare positive integer, and N a nonzero integer.
Titu Andreescu, Dorin Andrica
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This chapter gives the general theory and useful algorithms to find positive integer solutions (x, y) to general Pell’s equation (4.1.1), where D is a nonsquare positive integer, and N a nonzero integer.
Titu Andreescu, Dorin Andrica
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Particular Pell-Fermat Equations Revisited
2021See the abstract in the attached pdf.
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2018
For a quadratic field of discriminant D, the units of norm 1 are in one-to-one correspondence with the integer points of the equation X2 − DY2 = 4, often called the Pell equation. This makes it possible, as many throughout history have done, to study some aspects of quadratic fields through consideration of this associated Diophantine equation.
Samuel A. Hambleton, Hugh C. Williams
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For a quadratic field of discriminant D, the units of norm 1 are in one-to-one correspondence with the integer points of the equation X2 − DY2 = 4, often called the Pell equation. This makes it possible, as many throughout history have done, to study some aspects of quadratic fields through consideration of this associated Diophantine equation.
Samuel A. Hambleton, Hugh C. Williams
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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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The Mathematical Gazette, 2012
Given that it took until the 19th century for the case n = 5 of Fermat's last theorem to be settled, it is not surprising that Fermat's claim of having a proof for all exponents greater than 2 is nowadays treated with considerable scepticism.
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Given that it took until the 19th century for the case n = 5 of Fermat's last theorem to be settled, it is not surprising that Fermat's claim of having a proof for all exponents greater than 2 is nowadays treated with considerable scepticism.
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Periodica Mathematica Hungarica, 2015
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zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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