On norm form equations

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Parameterized norm form equations with arithmetic progressions

Buchmann and Pethő [5] observed that following algebraic integer 10 + 9α + 8α + 7α + 6α + 5α + 4α, with α = 3 is a unit. Since the coefficients form an arithmetic progressions they have found a solution to the Diophantine equation (1) NK/Q(x0 + αx1 + · · ·+ x6α) = ±1, such that (x0, . . . , x6) ∈ Z is an arithmetic progression. Recently Bérczes and Pethő [3] considered the Diophantine equation ...

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Norm Form Equations and Continued Fractions

We consider the Diophantine equation of the form x2−Dy2 = c, where c ∣∣ 2D, gcd(x, y) = 1, and provide criteria for solutions in terms of congruence conditions on the fundamental solution of the Pell Equation x2 − Dy2 = 1. The proofs are elementary, using only basic properties of simple continued fractions. The results generalize various criteria for such solutions, and expose the central norm,...

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On the number of solutions of norm form equations

A norm form is a form F(X" ... ,Xn ) with rational coefficients which factors into linear forms over C but is irreducible or a power of an irreducible form over Q. It is known that a nondegenerate norm form equation F(x" .... xn) = m has only finitely many. solutions (XI, .... Xn) E zn. We derive explicit bounds for the number of solutions. When F has coefficients in Z. these bounds depend only...

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A one-way function based on norm form equations

In this paper we present a new one-way function with collision resistance. The security of this function is based on the difficulty of solving a norm form equation. We prove that this function is collision resistant, so it can be used as a one-way hash function. We show that this construction probably provides a family of one-way functions.

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Solving Norm Form Equations Via Lattice Basis Reduction

The author uses irrationality and linear independence measures for certain algebraic numbers to derive explicit upper bounds for the solutions of related norm form equations. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm is then utilized to show that the integer solutions to NK/Q(x 4 √ N4 − 1 + y 4 √ N4 + 1 + z) = ±1 (where K = Q( 4 √ N4 − 1, 4 √ N4 + 1)) are given by (x, y, z) =...

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ژورنال

عنوان ژورنال: Journal of Number Theory

سال: 1977

ISSN: 0022-314X

DOI: 10.1016/0022-314x(77)90072-5