The Diophantine Equation B 2 X 4 ? Dy 2 = 1

” THE DIOPHANTINE EQUATION x n = Dy 2 + 1 ”

The Diophantine Equation b²X^4 - dY² = 1

If b and d are given positive integers with b > 1, then we show that the equation of the title possesses at most one solution in positive integers X, Y . Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with...

Thue ’ s theorem and the diophantine equation x 2 − Dy 2 = ± N KEITH MATTHEWS

A constructive version of a theorem of Thue is used to provide representations of certain integers as x2 −Dy2, where D = 2, 3, 5, 6, 7.

Some asymptotic limits for solutions of Burgers equation

§1 – Introduction. In this paper, we compute the limits (1) γ p = lim t → ∞ t 1 2 (1 − 1 p) u(·, t) p , 1 ≤ p ≤ ∞ for solutions u(·, t) of the equation (2) u t + a u x + b u u x = c u xx satisfying the Cauchy condition (3) u(x, 0) = u 0 (x), u 0 ∈ L 1 (R), that is, u(·, t) − u 0 1 → 0 as t → 0, t > 0. Here, u(·, t) p denotes the L p norm of u(·, t) as a function of x for fixed t, i.e., (4) u(·,...

THUE ’ S THEOREM AND THE DIOPHANTINE EQUATION x 2 − Dy

A constructive version of a theorem of Thue is used to provide representations of certain integers as x2 −Dy2, where D = 2, 3, 5, 6, 7.