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.

Three-dimensional elasticity solution for sandwich beams/wide plates with orthotropic phases: The negative discriminant case

In an earlier paper, Pagano (1969) [Pagano NJ. Exact solutions for composite laminates in cylindrical bending. J Compos. Mater. 1969; 3: 398–411] presented the three-dimensional elasticity solution for orthotropic beams (applicable also to sandwich beams) for the cases of: (1) a phase with positive discriminant of the qudratic characteristic equation, which is formed from the orthotropic material constants and further restricted to positive real roots and (2) an isotropic phase, which results in a zero discriminant. The roots in this case are all real, unequal, and positive (positive discriminant) or all real and equal (isotropic case). This purpose of this article is to present the corresponding solution for the cases of (1) negative discrimnant, in which case the two roots are complex conjugates and (2) positive discriminant but real negative roots. The case of negative discriminant is frequently encountered in sandwich construction, where the orthotropic core is stiffer in the transverse than the in-plane directions. Example problems with realistic materials are solved and compared with the classical and the first-order shear sandwich beam theories.

Three-Dimensional Elasticity Solution for Sandwich Plates With Orthotropic Phases: The Positive Discriminant Case

A three-dimensional elasticity solution for rectangular sandwich plates exists only under restrictive assumptions on the orthotropic material constants of the constitutive phases (i.e., face sheets and core). In particular, only for negative or zero discriminant of the cubic characteristic equation, which is formed from these constants (case of three real roots). The purpose of the present paper is to present the corresponding solution for the more challenging case of positive discriminant, in which two of the roots are complex conjugates.

Quadratic Non-Residues and Prime-Producing Polynomials

We will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.