The Two Chains

The Cunningham map c(n) = 2n+1 doubles and adds one. Starting from seeds 1 and 2, this single map generates all four odd primes of Z/2,310: {3, 5, 7, 11}. Then c(2*3) = 13 closes the boundary. Five prime factors, two seeds, one map.

The Two Cunningham Chains

A Cunningham chain of the first kind starts from a seed and repeatedly applies c(n) = 2n+1, continuing as long as each result is prime:

Interleaved by size: 1, 2, 3, 5, 7, 11. The two chains alternate perfectly. Every prime of Z/2,310 except 2 is generated by exactly one chain.

Cross-Chain Duality

Both chains stop when their OWN products appear. CC1(1) stops at 3*5 = 15. CC1(2) stops at 5*19 = 95 -- 5 appears again. 5 is the universal stopper. The shadow chain breaks at 13 where s(13) = 6 = 2*3 = composite.

The Shadow Function

The inverse of Cunningham: s(p) = (p-1)/2. Each prime shadows its predecessor:

The primes {2,3,5,7,11} are the longest initial prime segment where all shadows are prime or 1. At p=13, the shadow is 6 = 2*3 = composite. The chain breaks. Only 7 of 78,498 primes below 10^6 appear in shadow trees (0.009%). 5 divides 38.9% of CC1 stopping values (expected 20%, 1.95x enrichment).

The (p-1) Ladder

Every prime minus one equals 2 times its predecessor in the chain:

The Sum Identity

Why does the (p-1) column sum to 23 = CC1(2) at step 3? Four routes to the same number:

5 removes itself: sum(p-1) = sum(p) - 5. Everything minus 5 reveals the Cunningham chain. Dually: sum(p+1) = sum(p) + 5 = 28+5 = 33 = 3*11. The gap between the two sums = 2*5 = 10 = 5 seen from both sides.

The Shadow Sum

The shadow function s(p) = (p-1)/2 maps each odd prime to its predecessor:

s(3)=1, s(5)=2, s(7)=3, s(11)=5. Sum of shadows = 1+2+3+5 = 11.

Products tell the same story. Product(p-1) = 1*2*4*6*10 = 480 = 2^5*3*5. Product(p+1) = 3*4*6*8*12 = 6912 = 2^8*3^3. The neighborhoods are built from the same primes.

The Mirror Law

Mirror preservation: 210-1 = 209 and 420-1 = 419 both satisfy 2p = -1 (mod c(p)) across all four odd primes. The ring's key values carry the mirror law in their structure.

Cunningham-Mersenne Identity

Mersenne exponents n where M(n) is smooth in {2,3,5,7,11} are ONLY n in {1, 2, 3, 4, 6}:

The smooth exponents {1,2,3,4,6} are exactly the proper divisors of lambda(Z/210) = 2^2*3 = 12. The Cunningham chain, the Mersenne numbers, and the ring's lambda function are the same object seen from different angles.

The Smoothness Boundary

For each prime p, consider p^2 - 1 = (p-1)(p+1). For primes of Z/2,310, this product is always 11-smooth (all factors in {2,3,5,7,11}). How far does this pattern extend?

Recovery primes punctuate the boundary: 41 gives 1680 = 4*420, and 71 = c(5*7) gives 5040 = 7!. The smoothness extends past the ring's own primes, guarded by the Mersenne prime 31 and pierced at 37.

The Optimal Sieve

The chain generates the five primes of Z/2,310. Widen to all seven axiom primes {2, 3, 5, 7, 11, 13, 17} and a sharper fact appears: to strike out composites by trial division, these seven do it better than any other seven primes.

The sieve climbs the tower. Each level sieves with its primes; the smallest composite it MISSES is the square of the next prime:

All seven primes together sieve every composite up to 360 = 19^2 - 1 with no misses. The first thing they cannot catch is 361 = 19^2 -- the square of the eighth prime.

This boundary is also a limit on COMPUTATION. The seven channels see a number only through its seven residues, and each residue reports divisibility by its own prime. Every composite up to 360 has a prime factor among {2..17}, so at least one channel catches it. But 361 = 19^2 -- and any composite whose smallest prime factor exceeds 17 -- is nonzero in every channel, exactly like a prime. Past the sieve limit a prime and a large-factored composite become the SAME point in all seven channels, and no function of the residues can tell them apart. The optimal sieve and the limit of CRT-decomposable primality are one and the same bound.

Sieving is additive -- count what you strike out. The Euler product is its multiplicative twin, and it tells the same story.

As the exponent slides toward 1, the product becomes the Mertens product: 2/1 * 3/2 * 5/4 * 7/6 * 11/10 * 13/12 * 17/16 = 510,510 / 92,160. The numerator is the seven-primorial -- the radical of 214,414,200. The denominator is the product of (p-1) = 2^11 * 3^2 * 5, with no factor of 7.

CRT(23) Palindrome

23 = 2*11 + 1 is the first excluded Cunningham prime (CC1(2) step 3). Its CRT decomposition in Z/210 reads as a palindrome:

Cunningham of Products

Apply c(n)=2n+1 to products of primes from Z/2,310. Every prime result is structurally significant:

The c(5*7) = 71 identity: 71 is a prime whose p^2-1 equals 7 factorial. 5040 = 7! = 2^4*3^2*5*7, using four of five primes (only 11 absent). And 61+67 = 2^7 = 128.

The Cunningham Stopper

WHY do the chains stop? The multiplicative order of 2 modulo each odd prime reveals the structure. Every order uses only {2, 3, 5} -- the same three primes that build the chains:

2 is a primitive root (full period, ord = phi) mod {3, 5, 11, 13} -- these are the non-residue (NR) primes. 2 has half-period (ord = phi/2) mod {7, 17} -- the quadratic residue (QR) primes. The split encodes the deepest structure:

Explore: Chain Tracer

Enter a seed. Watch c(n) = 2n+1 build a Cunningham chain until it hits a composite. How far does YOUR seed go?

Trace Cunningham chain from seed:

Try: 1 (chain of length 3), 2 (chain of length 5), 89 (Sophie Germain chain), 6 (2*3 -> 13 on first step).

Contrast Table