摘要 :
The central role of the resolvent cubic in the solution of the quartic wasfirst appreciated by Leonard Euler (1707-1783). Euler's quartic solutionfirst appeared as a brief section (§ 5) in a paper on roots of equations [1, 2],and...
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The central role of the resolvent cubic in the solution of the quartic wasfirst appreciated by Leonard Euler (1707-1783). Euler's quartic solutionfirst appeared as a brief section (§ 5) in a paper on roots of equations [1, 2],and was later expanded into a chapter entitled 'Of a new method ofresolving equations of the fourth degree' (§§ 773-783) in his Elements ofalgebra [3, 4].
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摘要 :
There are very few quartic diophantine equations of the type f(x, y) = f (u, v), where f (x, y) = ax(4) + bx(3)y + cx(2)y(2) + dxy(3) + ey(4) is a binary quartic form, for which parametric solutions have been obtained. In this pap...
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There are very few quartic diophantine equations of the type f(x, y) = f (u, v), where f (x, y) = ax(4) + bx(3)y + cx(2)y(2) + dxy(3) + ey(4) is a binary quartic form, for which parametric solutions have been obtained. In this paper we obtain parametric solutions of such quartic equations when the coefficients a, b, c, d, e satisfy certain conditions.
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The generalized Gaussian distribution with location parameter , scale parameter sigma, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p = 1, 2, +, respectively. Derivations of...
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The generalized Gaussian distribution with location parameter , scale parameter sigma, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p = 1, 2, +, respectively. Derivations of the true maximum-likelihood estimators of and sigma for these special cases are popular exercises in many university courses. Here, we show how the true maximum-likelihood estimators of and sigma can be derived for p = 3, 4, 5. The derivations involve solving of quadratic, cubic, and quartic equations.
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While parametric solutions of the diophantine equation Sigma(i=1)(s) x(i)(4) = Sigma(i=1)(s) = y(i)(4) are known for any integral value of s greater than or equal to 2, the complete solution in integers is not known for any value ...
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While parametric solutions of the diophantine equation Sigma(i=1)(s) x(i)(4) = Sigma(i=1)(s) = y(i)(4) are known for any integral value of s greater than or equal to 2, the complete solution in integers is not known for any value of s. In this paper, we obtain the complete solution of this equation when s greater than or equal to 13. (C) 2004 Elsevier Inc. All rights reserved.
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In this paper, we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables Q(x(1), x(2), x(3), x(4)) = 0 can be expressed in terms of bilinear forms in four parameters. W...
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In this paper, we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables Q(x(1), x(2), x(3), x(4)) = 0 can be expressed in terms of bilinear forms in four parameters. We use this result to establish a necessary, though not sufficient, condition for the solvability of the simultaneous quadratic diophantine equations
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We propose a new method for solving a quartic equation, wherein the given quartic equation is transformed into a reciprocal equation using a Mobius transformation. We then use the property of reciprocal equations to solve the quartic equation.
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We prove that diophantine equation in the title has at most one positive integer solution for any positive integers A>1, B>1. It follows that Lucas problem is very simple to solve and a recent result of Bennett is very simple to prove.
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For b an odd integer whose square-free part has at most two prime divisors, it is shown that the equations in the title have a common solution in positive integers precisely when b divides 4a(2) - 1 and the quotient is a perfect s...
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For b an odd integer whose square-free part has at most two prime divisors, it is shown that the equations in the title have a common solution in positive integers precisely when b divides 4a(2) - 1 and the quotient is a perfect square. The proof provides an explicit formula for the common solution, known to be unique. Similar results are obtained assuming the square-free part of b is even or has three prime divisors.
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This paper presents a general analysis of all the quartic equations with real coefficients and multiple roots; this analysis revealed some unknown formulae to solve each kind of these equations and some precisions about the relati...
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This paper presents a general analysis of all the quartic equations with real coefficients and multiple roots; this analysis revealed some unknown formulae to solve each kind of these equations and some precisions about the relation between these ones and the Resolvent Cubic; for example, it is well-known that any quartic equation has multiple roots whenever its Resolvent Cubic also has multiple roots; however, this analysis reveals that any non-biquadratic quartic equation and its Resolvent Cubic always have the same number of multiple roots; additionally, the four roots of any quartic equation with multiple roots are real whenever some specific forms of its Resolvent Cubic have three non-negative real roots. This analysis also proves that any method to solve third-degree equations is unnecessary to solve quartic equations with multiple roots, despite the existence of the Resolvent Cubic; finally, here is developed a generalized variation of the Ferrari Method and the Descartes Method, which help to avoid complex arithmetic operations during the resolution of any quartic equation with real coefficients, even though this equation has non-real roots; and a new, more simplified form of the discriminant of the quartic equations is also featured here.
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