A New Look at the Single Ladder Problem (SLP) via Integer Parametric Solutions to the Corresponding Quartic Equation

Abstract

The aim is to put new light on the single ladder problem (SLP). Some new methods for finding complete integer solutions to the corresponding quartic equation z 4 −2L z 3 +( L 2 − a 2 − b 2 ) z 2 +2L a 2 z− L 2 a 2 =0 z4−2Lz3+(L2−a2−b2)z2+2La2z−L2a2=0 are developed. For the case L≥ L min L≥Lmin , these methods imply a complete parametric representation for integer solutions of SLP in the first quadrant. Some corresponding (less complete) results for the case L> L min L>Lmin are also pointed out. Keywords: single ladder problem (SLP); integer parametric solutions; simultaneous quadratic equations; quartic equations; algebraic equations; recreational mathematics