From the slope of the dome to its feet on themodulo-30 wheel

In our paper on the dome of a prime [1] every semiprime N = pq is the right angle ofan isosceles triangle in the plane (s, d), s = (p + q)/2, d = (q − p)/2, whose feet are thesquares p2 and q2. The present paper treats the inverse problem: given only N, reach thefeet. The shape of the triangle is fixed by one angle u, e2u = q/p, and the slope of thedome, coth u, is taken as the unknown. A trial slope coth v, e2v = µ/ν, corresponds to aneasy dome, the dome of µνN = (µp)(νq), whose apex x = µp + νq lies at the exact distance2√µνN (cosh(u − v) − 1) above √4µνN; a hyperbolic rotation returns the hard dome. Tworesults make this a method of the wheel. First, the wheel fixes the apex of every easy domebefore any test: x ≡ µRp + νRq (mod 30) for the pairs of vectors with RpRq ≡ N, atmost eight classes out of thirty, which are the classes of s and d of the base table carried toµνN. Second, the identity of the easy dome together with Dirichlet’s approximation proveson the dome the bound of Lehman: if no prime factor lies below N1/3, some fraction withµν ≤ N1/3 + 1 reaches the feet within N1/6/(4√µν) steps, so the complete search eitherfactors N or proves it prime. The search is implemented in standard C without threads orexternal libraries (wheel easy.c). On one machine with two cores it factored, blind withinthe whole hard band 1 < q/p ≤ 3.695, all 46 hard semiprimes of a test set of 25 to 33digits (at most 340 s), 15 with random ratios, the 12 test semiprimes of 35 and 40 digits ofPaper VII, the 17 hard semiprimes of the 30-digit window of Paper VII (432 s in all), threesemiprimes of 35 digits, two of them blind (221, 198 and 77 s, the last one not factored within9 minutes by the earlier search) and a blind semiprime of 40 digits, searched in the hard layerin which it was drawn (20.5 min). Against the jumps of the apex and the slope windows onthe same machine the gain is a factor of 8 to 1,260 at 29–31 digits. An estimate of q/p torelative accuracy η divides the work by 0.13/η, and ratios close to simple fractions are foundin milliseconds for semiprimes of any size. The order is N1/3, and the subexponential sievesremain the tool for large semiprimes with random factors.