The Landau Problem Family: Twin Primes, Legendre's Conjecture and Primes of the Form 𝑛² + 1 as They Flow from the Goldbach Lattice in Quaternionic Eulerian Calculus

At the 1912 International Congress of Mathematicians Edmund Landau named four problems about primes: Goldbach's conjecture, the twin prime conjecture, Legendre's conjecture, and the infinitude of primes of the form n² + 1. In the author's treatment of Goldbach's conjecture in Quaternionic Eulerian Calculus (QEC), every even number 2n stands at the corner (n, n) of the line of reflection y = x, the confluence n + n of two QECian primes, and is the midpoint of its own anti-diagonal x + y = 2n. The same corner is the corner of an n × n square of area n², and by the QEC difference quotient taken at exactly zero, the Goldbach number 2n is the derivative of that area. This paper shows that the other three Landau problems live on the same corner. A twin pair (n − 1, n + 1) is the Goldbach pair of 2n at the closest unequal spacing, and its product is n² − 1, one unit below the square corner. The Legendre gap between consecutive square corners has length (n + 1)² − n² = 2n + 1 = n + (n + 1), exactly the odd number that the line of reflection places between the corners (n, n) and (n + 1, n + 1). And n² + 1, one unit above the square corner, is the quadrance of the prime-extractor triangle of base 1 and slope n. For each problem the geometric reduction is proved in full, together with the demonstrated results that bear on it: every twin pair beyond (3, 5) balances on a multiple of 6; every composite in a Legendre gap is struck by a prime no larger than n; every odd prime dividing n² + 1 is ≡ 1 (mod 4), and from this there are infinitely many such primes. Each problem is then completed by the QEC principle of inheritance of the rule, the same principle that completes Goldbach's conjecture, applied to its own place on the corner. All quantities are pure numbers.