independent run: pairs with k = 12mu, mu <= 600: 4040, R histogram {1: 3306, 2: 703, 3: 31}
independent run: pairs with even k <= 2400, 12 !| k: 5393, R histogram {1: 3795, 2: 1199, 3: 399}
total: 9433
smallest k = 12mu with a Case-(2) prime: 84 primes: [11, 17]
min p: 11  p = 13 occurs: False  all p <= 3mu: True  max p/mu: (2.4285714285714284, 84, 17)
all k' <= k/p and k' = k mod (p-1): True
first k = 12mu with R >= 2: k = 504 (mu = 42), p = 11
first k = 12mu with R >= 3: k = 5364 (mu = 447), p = 11
Table 1: k=84 p=11: k'_min=4 R=1 chain 84 > 4  (chain length 1 == R: True)
Table 1: k=84 p=17: k'_min=4 R=1 chain 84 > 4  (chain length 1 == R: True)
Table 1: k=96 p=11: k'_min=6 R=1 chain 96 > 6  (chain length 1 == R: True)
Table 1: k=132 p=11: k'_min=12 R=1 chain 132 > 12  (chain length 1 == R: True)
Table 1: k=252 p=11: k'_min=12 R=1 chain 252 > 12  (chain length 1 == R: True)
Table 1: k=504 p=11: k'_min=44 R=2 chain 504 > 44 > 4  (chain length 2 == R: True)
Table 1: k=5544 p=11: k'_min=484 R=3 chain 5544 > 484 > 44 > 4  (chain length 3 == R: True)
Table 1: k=5544 p=1109: k'_min=4 R=1 chain 5544 > 4  (chain length 1 == R: True)
primary run: ## k=12mu, mu<=300: weights=300  Case(1) pairs=3077  Case(2) pairs=1631  Case(3) pairs (primes <= max(k+1,N))=76982  unresolved=0  filtration mismatches=0  r_max histogram={1: 1394, 2: 237}
primary run: ## Case(2) pairs NOT covered by the source's Theorem 2 (k < p*k2): 240; first 40 (k, p, k'_min, k2): [(252, 11, 12, 32), (384, 11, 24, 44), (396, 17, 12, 28), (444, 19, 12, 30), (516, 11, 36, 56), (540, 23, 12, 34), (564, 11, 44, 54), (600, 17, 24, 40), (648, 11, 48, 68), (672, 19, 24, 42), (684, 11
primary run: ## source's Theorem 3 check (p>=13 | 2k(2k+2)(2k+4)(2k+6)(2k+8)(2k+12)): 764 pairs where 'Case (2) and p | a_(mu+1)' fails literally; all of them are Case-(1) primes ((p-1)|k): True; first Case-(1) exceptions (k,p): [(12, 13), (24, 13), (36, 13), (36, 19), (36, 37), (48, 13), (48, 17), (60, 31), (60
