The wave of a prime on the modulo-30 wheelExact counts of its points, of its semiprimes and of its crossings with other waves,separated from the statistical laws

In our series on the modulo-30 wheel every prime p ≥ 7 is a wave on the eight vectorsVR = R + 30n, and every semiprime N = pq is the crossing of the wave of p with the waveof q. This note collects what can be said about one wave and about the crossings of severalwaves, and separates the statements into three kinds. Exact algebraic statements, proved forevery bound X: the number of points of the wave of p below X is Φ(X/p) − Φ(p − 1), withΦ the count of integers prime to 30, given in closed form; the number of its semiprimes isexactly π(X/p) − π(p − 1), distributed over the vectors by the classes of the cofactor; theposition of pq on its vector is p nq + c with an explicit constant c; the wave occupies thepositions n ≡ −R·30−1(mod p) of VR; two waves p < r cross exactly Φ(X/(pr)) times belowX, at positions given by the Chinese remainder theorem, and exactly one of these crossings,pr itself, is a semiprime; the points that the wave of p reaches first are counted by a Legendreformula, and they are all semiprimes as soon as p3 > X. Proved asymptotic statements: theprime number theorem, Dirichlet’s theorem and Mertens’ theorem give the size of each countand the equal share 18of every vector. Statistical and conjectural statements: the number ofsemiprimes on a fixed diagonal or crossing diagonal follows the Hardy–Littlewood weights,and that none of these lines is empty is the prime-pair and the Goldbach conjecture. Allexact statements were checked against an independent sieve for every one of the 443 wavesbelow 107, without exception, by a program that uses no external library. The waves of 2,3 and 5 obey the same position and angle laws on the 22 struck-out positions, onto whichthey map the eight vectors one to one, two to one and four to one. In a period of ten rowsof the base table two entries share a wave only if its prime divides their distance, so no waveabove 294 links two entries; the value s = (p + q)/2 of a semiprime of the period lies inan explicit layer of width 147/p, its parity is fixed by N mod 4, and residue filters from thewheel reduce the square tests of Fermat’s method, for N = 299939 from 95 to one. Thehalving–reflection map R 7→ R/2 or N − R, started at (N − 1)/2, always reaches a primethat never divides N, and its cycle has exactly h(N) halvings, the order of 2 modulo ±1,from which s = (N + 1 − mh(N))/2 and d follow for a semiprime, at the cost of walking thecycle; the cycle also yields 4s ≡ 2N+1 (mod N), which a baby-step giant-step search insidethe layer of s, cut by the parity of s and its class modulo 3, solves in about N1/4 operationsfor close factors, e.g. 13,832 operations for a 20-digit semiprime; an adaptive search fromboth ends of the layers is up to 30 times faster for balanced semiprimes, and slower by up to2.5 for the middle type. Perfect squares on the cycle, tested as they appear and combinedby linear algebra over GF(2), give X2 ≡ 1 (mod N) and factor hard 20-digit semiprimesafter about 3 × 106steps of the cycle; in the hard region q/p ≈ 3 their cost equals that ofrandom semiprimes, and starts at perfect squares on the fifth or eighth vector give the samecost within half a percent. Replacing the reflection N − R by the addition R + N removesthe sign, halves the steps of the single squares, and lets a program in C test the values of thecycle by residues without division; it factors the seven 30- and 31-digit semiprimes of PapersII and IV left open by the search, with q/p from 3.2 to 4000, in 8 to 21 minutes each on twocores. On the helix of the wheel the wave of 7 multiplies every angle by 7; the theorem givesevery point of this helix and every crossing of it, but not which crossing carries a given N,which is the factoring problem itself.