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Starting in the late 1960s computer scientists including Dijkstra and Hoare advocated goal-oriented programming and the formal derivation of algorithms. The chief impediment to realizing this for loop-based programs was that a priori determination of loop-invariants, a prerequisite for developing loops, was a task too complex for any but the simplest of operations. Around 2000, these techniques were...
Using the formal derivative idea, we give a generalization for the Cauchy’s Theorem relating to the factors of (x + y)n−xn− yn. We determine the polynomials A(n, a, b) and B(n, a, b) such that the polynomial $$ A(n, a, b)(x + y)^n + B(n, a, b)(x^n + y^n) $$ can be expanded, for any natural number n, in terms of the polynomials x+y and ax2+bxy + ay2. We show that the coefficients of...
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