The Last Smooth Pair

2400 = 2^5*3*5^2, 2401 = 7^4

Stormer's theorem (1897): finitely many consecutive pairs where both are 11-smooth. The largest such pair is (2400, 2401). Their CRT decompositions reveal zero-trading: the mod-8 and mod-25 channels are zero on one side, the mod-49 channel is zero on the other.

The Stormer Zero-Trading Theorem

2400 = 2^5 * 3 * 5^2. 2401 = 7^4. Their CRT decompositions reveal which channels carry zeros:

Channel24002401Trade
mod 80 (ZERO)12 exits
mod 96 = 2*37neutral
mod 250 (ZERO)15 exits
mod 4948 = 7^2-10 (ZERO)7 enters
mod 1123neutral

2400 has zeros in the mod-8 and mod-25 channels. 2401 has zero in the mod-49 channel. The zeros trade: 2 and 5 yield to 7.

Channel Anatomy

2400 mod 9
6 = 2*3
The product of the two smallest primes sits in the mod-9 slot.
2400 mod 49
48 = 7^2 - 1
One less than the modulus. Full capacity.
2401 mod 8
1
Identity. Reset.
2401 mod 9
7
7 appears in the mod-9 channel.

When 2400 uses primes 2 and 5, it fills the mod-49 channel to 48 = 7^2 - 1. One step later, 7 takes over: the mod-49 channel drops to 0, and the mod-8 and mod-25 channels reset to 1.

The Stormer Sum Theorem

Stormer Sum (PROVED)
2400 + 2401 = 4801 is prime. CRT(4801) = (1, 4, 1, 48, 5). The traded channels (mod 8, mod 25) show 1 = identity = neutralized. The mod-49 channel shows 48 = 7^2 - 1. The mod-11 channel shows 5. The sum remembers which channels traded.

Census of Smooth Pairs

Every consecutive smooth pair trades zeros (since gcd(n,n+1) = 1, their prime factors are disjoint). The larger pairs show increasingly dramatic trades:

nn+1FactorizationZero Trade
892^3, 3^22 -> 3
48492^4*3, 7^22*3 -> 7
80812^4*5, 3^42*5 -> 3
1201212^3*3*5, 11^22*3*5 -> 11
2242252^5*7, 3^2*5^22*7 -> 3*5
3843852^7*3, 5*7*112*3 -> 5*7*11
240024012^5*3*5^2, 7^42*3*5 -> 7 (final)

The final pair is the most extreme: three primes on one side, one on the other. 7 absorbs 2, 3, and 5. Only 11 stays neutral (never zeros in either).

Why 7^4?

In the last smooth pair, 2401 = 7^4 is a pure power of a single prime. The other primes (2, 3, 5) all appear in 2400. 11 divides neither.

Identity
7^4 - 1 = 2^5*3*5^2
Three primes fit exactly inside 7^4 - 1 = 2400.
Why not 7^3?
7^3 + 1 = 344 = 8*43, not smooth
43 is a Heegner number. Only at 7^4 does 7 find a smooth neighbor.
f(7) = 41
p^2 - p - 1 evaluated at p = 7
41 is prime. The Heegner numbers control where smoothness dies.

Contrast

AspectStandard viewRing structure
The pair2400 and 2401 are consecutive integers. Stormer 1897 curiosityCRT zeros trade: mod-8 and mod-25 yield to mod-49. Structural
Why last?Analytic bound from Pell equations7^3+1 blocked by Heegner 43. Only 7^4 finds a smooth neighbor
Sum4801 is prime, no further significanceCRT shows traded channels neutralized to 1. Sum remembers the trade
11 channel11 divides neither -- unremarkable11 stays neutral: never participates in the zero-trades

Explore: Zero-Trade Explorer

Enter n to factor (n, n+1) and see their CRT decompositions. Watch which channels carry zeros and how they trade.

Explore pair starting at n:

Try: 8 (2^3), 48 (2^4*3), 80 (2^4*5), 224 (2^5*7), 2400 (the last smooth pair).

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