(2400, 2401) = the largest consecutive pair using only axiom primes
2400 and 2401 are consecutive integers. Stormer's theorem (1897) says finitely many consecutive pairs are B-smooth for any prime bound B. A curiosity in analytic number theory.
The last smooth pair TRADES zeros between CRT channels. D*E channels empty on one side; b channel empties on the other. Smoothness ends when depth absorbs everything. The boundary is structural, not accidental.
An integer is 11-smooth if its only prime factors come from {2, 3, 5, 7, 11} = the axiom primes. Stormer's theorem (1897): there are finitely many consecutive pairs where both are 11-smooth. The LARGEST such pair is:
Their CRT decompositions reveal the zero-trading pattern:
| Channel | mod | 2400 | 2401 | Trade |
|---|---|---|---|---|
| D (mod 8) | 8 | 0 | 1 = sigma | D exits |
| K (mod 9) | 9 | 6 = D*K | 7 = b | - |
| E (mod 25) | 25 | 0 | 1 = sigma | E exits |
| b (mod 49) | 49 | 48 = b2-1 | 0 | b enters |
| L (mod 11) | 11 | 2 = D | 3 = K | - |
2400 has zeros in the D and E channels. 2401 has zero in the b channel. The zeros TRADE: D * E yield to b. Bridge and observer step aside. Depth stands alone.
The non-zero residues are axiom-structured:
K-channel = 6 = D*K (the thorn). b-channel = 48 = b2 - 1 = mirror of sigma in Z/49. L-channel = 2 = D (bridge echoes in the protector).
D-channel = 1 = sigma (ground state). K-channel = 7 = b (depth appears in the closure channel!). E-channel = 1 = sigma. L-channel = 3 = K (closure echoes in the protector).
When 2400 occupies D and E, it fills b's channel to the brim: 48 = b2 - 1. One step later, b takes over: its channel drops to 0, and D and E reset to sigma = 1.
The sum of the last smooth pair is prime. Its CRT decomposition:
| D (mod 8) | K (mod 9) | E (mod 25) | b (mod 49) | L (mod 11) |
|---|---|---|---|---|
| 1 = sigma | 4 = D2 | 1 = sigma | 48 = b2-1 | 5 = E |
The traded channels (D, E) show sigma = ground state = neutralized. The b-channel shows 48 = b2 - 1 = mirror of sigma. The protector shows E = the observer. The sum remembers which channels traded.
Every consecutive smooth pair trades zeros (since gcd(n, n+1) = 1, their prime factors are disjoint). The larger pairs show increasingly dramatic trades:
| n | n+1 | Factorization | Zero Trade |
|---|---|---|---|
| 8 | 9 | D3, K2 | D -> K |
| 10 | 11 | D*E, L | D*E -> L |
| 48 | 49 | D4*K, b2 | D*K -> b |
| 49 | 50 | b2, D*E2 | b -> D*E |
| 80 | 81 | D4*E, K4 | D*E -> K |
| 120 | 121 | D3*K*E, L2 | D*K*E -> L |
| 224 | 225 | D5*b, K2*E2 | D*b -> K*E |
| 242 | 243 | D*L2, K5 | D*L -> K |
| 384 | 385 | D7*K, E*b*L | D*K -> E*b*L |
| 440 | 441 | D3*E*L, K2*b2 | D*E*L -> K*b |
| 539 | 540 | b2*L, D2*K3*E | b*L -> D*K*E |
| 2400 | 2401 | D5*K*E2, b4 | D*K*E -> b |
The final pair is the most extreme: three primes on one side, one on the other. Depth absorbs bridge, closure, and observer. Only the protector L stays neutral (never zeros in either).
In the last smooth pair, 2401 = b4 is a pure power of a SINGLE axiom prime. All other axiom primes (D, K, E) appear in 2400. The protector L divides neither.
Rewritten: b4 - 1 = D5 * K * E2 = 2400. The entire axiom minus depth fits inside b4 - 1.
Why? b3 = 343, but 344 = 8 * 43, and 43 is a Heegner number (intruder prime). b3 + 1 is NOT smooth. Only at b4 does depth find a smooth neighbor: b4 - 1 = 2400.
The depth quadratic f(b) = 41 = KEY, and f(K) + D = b (Heegner connection). Depth and the Heegner numbers control where smoothness dies.
CRT channel values for 2400 (blue) and 2401 (green). Zeros highlighted in red.
Enter a number to check if it and its neighbor are both 11-smooth:
The Stormer boundary connects to:
The Universal Boundary - 8 intruder primes that block smoothness. 43 blocks b3+1.
The Depth Quadratic - f(b)=41=KEY. The depth quadratic controls the Stormer boundary.
The Nine Heegner Numbers - 43 is a Heegner number. It guards the depth boundary.
Why 37 Comes Home - 37=p_12, the prodigal prime. CRT(37)=(E,sigma,D2K,37,D2).
The Two Chains - Cunningham chains generate the axiom. The smooth boundary is where chains stop.
Shadow Evaluations - The Stormer pair (2400,2401) proves exactly L=11 shadow polynomial values are smooth.
Stormer's theorem (1897). CRT zero-trading: proved S709. Stormer Sum Theorem (4801 prime): proved S711. 130 interactive pages. Return to hub