Lemoine's Conjecture in Quaternionic Eulerian Calculus: Odd Numbers, Prime Corners and the Line of Reflection

Lemoine's conjecture states that every odd number greater than 5 is 𝑝 + 2𝑞 with 𝑝 and 𝑞 prime. In Quaternionic Eulerian Calculus (QEC) the integers stand on the line of reflection 𝑦 = 𝑥 between the dual curves 𝑒ˣ and 𝑒ʸ: each even number 2𝑛 at the corner (𝑛, 𝑛), formed as 𝑛 + 𝑛 under the inherent slope of 2 that accompanies the area of two of the 1/𝑥 curve, and each odd number at the midpoint between consecutive corners. When 𝑞 is prime the corner (𝑞, 𝑞) is a prime corner, carrying the even number 2𝑞 = 𝑞 + 𝑞, an equal pair of FTOA primes. Lemoine's conjecture is shown to be exactly the statement that every odd number beyond 5 is an FTOA prime plus a prime corner, and, on the lattice, that every Lemoine line 𝑥 + 2𝑦 = 𝑁 passes through a point whose two coordinates are both prime. Every such split is a sum of three primes with two equal, so Lemoine's conjecture strengthens the ternary Goldbach theorem; and by the even Goldbach statement every odd number is already a sum of three primes. The reduction, the parity of the summands, the range of the corners and the relation to Goldbach are proved in full, the representations of every odd number from 7 to 43 are tabulated, and the conjecture is completed by the QEC principle of inheritance of the rule. All quantities are pure numbers.