The s–d base table of the modulo-30 wheel

Every entry of the modulo-30 wheel is expressed through the offsets ρ ∈ {0, 4, 6, 10, 12, 16, 22, 24}of its eight vectors, and every product of two entries as N = s2 − d2, where s = 7 + α + 15kand d = β + 15j are linear in the crossing diagonal k = m + n and the diagonal j = n − m ofthe two factors (formulas of the first author). On these coordinates a base table of 36 grids isconstructed, one for each family F(a, b), in which every product of two wheel entries occupiesexactly one cell that yields p, q, s, d and N simultaneously. The parametrisation, the familiesof each vector, the position law, an ordered form with constant second difference 1800, thelayers of the position, the diagonal element associated with each N, and a formulation ofthe complete system in the single unknown k are established. For the hard semiprimes thediagonals are shown to appear in periods of increasing length, Bt+1 = Bt + 2C∆L2; thehalf-difference satisfies d ≤ 3.840√n (verified to 1014), and an exact generator of all hardsemiprimes in increasing order is obtained. For a given N the factorisation is the uniquelattice point of the hyperbola s2 − d2 = N in its grid. It is located by parallel streams, byrotating residue matrices, and by a baby-step giant-step search in k based on the congruence4s ≡ 2N+1 (mod N); a single search shared by all families lowers the worst-case cost by afactor between 2 and 2.83, and 57 hard semiprimes of 20 to 40 digits are factored. Threewindow methods are introduced: an adaptive wheel table whose length, a multiple of aprimorial, is placed near the expected smaller factor (70 digits in 34 seconds); a dynamicsieve on a growing wheel 30, 210, 2310, . . . that retains the eight levels of the wheel 30 andclassifies a window completely without a probabilistic test, with up to 1.73 times fewerfactors than the wheel 30; and the base table in one variable, E = X +Dk + 30n, coloured bydirectional multiplication, whose bit-level implementation is 3.7 times faster. A window of30,000 integers at 9 · 1029 is classified completely: its 425 primes and its 316 semiprimes withboth factors above 1010 are determined deterministically, in agreement with the expecteddensities in every period. The method determines an interval rather than a single numberand runs in independent periods on ordinary processors. A hard semiprime of 24 digits isfactored on the same table in 504 seconds.