The Goldbach Conjecture in Quaternionic Eulerian Calculus: The 1/x Curve, the Area of Two, and the Line of Reflection

Goldbach's conjecture states that every even number greater than 2 is the sum of two primes. This paper treats it in Quaternionic Eulerian Calculus (QEC), the author's calculus built on A. L. Baker's quaternion algebra, in which a number is a ratio, a ratio is a slope, and a slope is a derivative. The argument begins on the curve 𝑦 = 1/𝑥. Every tangent to it cuts from the axes a right triangle of area exactly 2, and by the Eulerian fundamental theorem of calculus, under which derivative and integral are indistinguishable at the instantaneous centre of revolution, that constant area of 2 carries a constant inherent slope of 2: Area = [(2/𝑥) · 𝑥] ≡ Slope = [2𝑥/𝑥] = 2[(𝑦 = 𝑥)/𝑥] = 2. With 𝑥 and 𝑦 taken as positive integers, 2𝑥 runs through every even number, and the inherent slope holds 𝑦 equal to 𝑥 on the line of reflection 𝑦 = 𝑥 between the dual exponential curves 𝑒ˣ and 𝑒ʸ. There every integer is divisible only by 1 and itself positionally: a QECian prime. Every even number 2𝑛 is therefore the confluence 𝑛 + 𝑛 of two QECian primes at the corner (𝑛, 𝑛), and the midpoint of its own anti-diagonal 𝑥 + 𝑦 = 2𝑛, whose lattice points are the symmetric pairs (𝑛 − 𝑘, 𝑛 + 𝑘). Goldbach's conjecture is shown to be exactly the statement that every such midpoint balances a pair of FTOA primes, the primes of the fundamental theorem of arithmetic. The FTOA primes are shown to lie among the QECian primes, and the passage from QECian pairs to FTOA pairs is the QEC principle of inheritance of the rule, from which the conjecture follows. All quantities are pure numbers.