An Elementary Algorithm for Solving a Diophantine Equation of Degree Fourth with Runge’s Condition

We propose an elementary algorithm for solving the diophantine equation (p(x; y) + a1x + b1y)(p(x; y) + a2x + b2y)- dp(x; y)- a3x - b3y -c = 0 ( *) of degree fourth, where p(x; y) denotes an irreducible quadratic form of positive discriminant and (a1; b1) ̸= (a2; b2). The last condition guarantees that the equation ( ) can be solved using the well known Runge’s method, but we prefer to avoid the use of any power series that leads to upper bounds for solutions useless for a computer implementation.

ESTIMATES FOR THE SOLUTIONS OF CERTAIN DIOPHANTINE EQUATIONS BY RUNGE'S METHOD

We consider the two Diophantine equations ym = F(x) and G(y) = F(x) under the assumption that gcd (m, deg F) > 1 and gcd ( deg G, deg F) > 1, respectively. We prove that the bounds for the denominator of the coefficients of the power series arising from the above two situations can be improved considerably and thus we establish improved upper bounds for the size of the solutions (namely for |x| and |y|). We also give explicit upper bounds for the integer solutions of equations of the form [Formula: see text] under the assumption that [Formula: see text]

Runge Solution of the Continuum Equations for Spherical Electrostatic Plasma Probes

An approximate but convergent method is used to determine the electric potential and charged particle density near non-small spherical probes in a weakly ionized continuum plasma. Quantities determined partly by experiment (e, PP' yp) and partly from continuum theory (J +, J _) are introduced into three continuum differential equations which are then solved using Runge's method. The initial conditions must be estimated iteratively until the sheath solution joins smoothly to the quasi-neutral solution. Two different types of solution curves are discussed.