One polynomial. Its zeros are the golden ratio and -1/phi. Its discriminant is 5. It maps every prime in {3, 5, 7, 11} to a prime, and the pattern breaks at 13. f(p) = -1 mod p for ALL p -- this polynomial always equals the mirror element in its own residue ring. And f(7) = 41: add 1 and you get 42.
Values at the Ring's Primes
p
f(p)
Prime?
Primitive root?
2
1
--
--
3
5
YES
ord(3,5) = 4 = phi(5). Full!
5
19
YES
ord(5,19) = 9 = phi/2. Half!
7
41
YES
ord(7,41) = 40 = phi(41). Full!
11
109
YES
ord(11,109) = 108 = phi(109). Full!
13
155 = 5*31
NO
WALL
f(p) = -1 (mod p) for ALL p. Always. The polynomial equals the mirror element (p-1) in Z/p for every prime. Four consecutive odd primes (3, 5, 7, 11) give prime outputs. The composite wall at 13 is the same wall that stops the Cunningham chain.
Six Roles
1. MIRROR
f(p) = -1 mod p
Always the mirror. Universal.
2. NORM
f(p) = Norm(p - phi)
Golden ratio. Discriminant = 5.
3. ARTIN
delta(7) = C*(1 - 1/41)
41 = f(7) governs Artin's conjecture for 7.
4. FORBIDDEN
f(7) = 41
41 = forbidden index for 7 as primitive root.
5. POWER
attenuation
Power function attenuation through f(p).
6. QR-PARITY
(p/f(p)) = (-1/p)
Legendre parity transmission.
The f(p) + 1 Identity
f(n) + 1 = n(n-1). This links polynomial outputs to ring constants:
p
f(p)
f(p)+1
f(p)+2
3
5
6 = 2*3
7 (Heegner!)
5
19
20 = 2^2*5
21 = 3*7
7
41
42 = 2*3*7
43 (Heegner!)
11
109
110 = 2*5*11
111 = 3*37
f(7) + 1 = 42 = 2*3*7. f(3)+2 = 7 and f(7)+2 = 43 are both Heegner numbers -- the polynomial connects to class number 1 fields.
Sum of f(p) Values
Sum Theorem (PROVED)
f(3) + f(5) + f(7) + f(11) = 5 + 19 + 41 + 109 = 174 = 2*3*29 = 6*29. The sum of all four outputs = 6 * 29, where 29 = 2^2 + 5^2 = 4 + 25.
Z/12,612,600 vs Z/210
Ring Difference Theorem (PROVED)
12,612,600 - 210 = 12,612,390 = 210 * 19 * 109 * 29 = 210 * f(5) * f(11) * 29. The two rings are indistinguishable through f(5) and f(11), but differ mod f(7) = 41: mod 41, 12,612,600 gives 16 while 210 gives 5. Only f(7) distinguishes them.
Stormer Zero-Trading
The largest consecutive 11-smooth pair: (2400, 2401) = (2^5*3*5^2, 7^4).
2400 = 2^5*3*5^2
CRT: (0, 6, 0, 48, 2)
mod-8 and mod-25 channels = 0.
2401 = 7^4
CRT: (1, 7, 1, 0, 3)
mod-49 channel = 0.
The last 11-smooth pair trades zeros across CRT channels: 2400 zeroes the mod-8 and mod-25 channels, while 2401 = 7^4 zeroes the mod-49 channel. Beyond this pair, no consecutive integers are 11-smooth.
Shadow Polynomial Evaluations
x
P(x)
Factored
Meaning
0
30
2*3*5
Constant term
6
60
2^2*3*5
lambda(Z/2,310)
7
240
2^4*3*5
|roots(E8)|
9
1344
8*168 = 2^3*|PSL(2,7)|
Fano-PSL
11
4320
2^5*3^3*5
P(0)*lambda(Z/210)^2
13
10560
2^5*3*5*11
All 5 primes
P(7) = 240, which is the number of roots in the E8 lattice. rank(E8) = 8 = P(7)/P(0). dim(E8) = 240 + 8 = 248. The shadow polynomial at x = 7 reproduces E8 geometry.
CRT Root Anatomy
Where does f(p) = p^2-p-1 have roots in the Z/214,414,200 CRT channels? Discriminant 5 governs: roots exist mod q iff Legendre (5/q) = +1.
Channel
Modulus
Roots
Reason
mod 8 (Z/8)
Z/8
0
f(x) always odd -- 2 never divides
mod 9 (Z/9)
Z/9
0
(5/3) = -1
mod 25 (Z/25)
Z/25
0
Hensel failure: f(3+5t) = 5 mod 25
mod 49 (Z/49)
Z/49
0
(5/7) = -1
mod 11 (Z/11)
Z/11
2
(5/11) = +1. UNIQUE!
mod 13 (Z/13)
Z/13
0
(5/13) = -1
mod 17 (Z/17)
Z/17
0
(5/17) = -1
11 is the UNIQUE channel. The two roots are 4 and 8 (= 2^2 and 2^3) -- the golden ratio and its conjugate, mod 11. Sum 4+8 = 12 = 1 mod 11 (Vieta). Product 4*8 = 32 = -1 mod 11 (mirror). The golden ratio lives in exactly one of seven channels.
Golden roots
phi = 8 mod 11, 1-phi = 4 mod 11
The golden ratio hides in the mod-11 channel.
mod-25: no roots
25 has zero roots (Hensel failure)
Double root at 3 mod 5 lifts to residue 5 mod 25, never 0. Discriminant 5 blocks its own channel.
Smooth returns
f(2)=1, f(3)=5, f(37)=11^3
Exactly 3 smooth values. The 37th = 1331.
CRT Root Theorem (PROVED)
f(p) = p^2-p-1 has discriminant 5. Among the 7 CRT channels of Z/214,414,200, mod 11 is the unique channel with roots: 4 and 8 (= 2^2 and 2^3). Mechanism: Legendre (5/p) = +1 only at p = 11. mod 25 has zero roots (Hensel failure at double root). f(p) is 11-smooth for exactly 3 prime inputs: f(2) = 1, f(3) = 5, f(37) = 11^3 = 1331.
Explore: f(p) Calculator
Enter any number n. See f(n) = n^2 - n - 1, whether it is prime, and the centered square CS(n) = 2*f(n)+3.
A quadratic with discriminant 5 and golden ratio zeros. Well-studied in algebraic number theory.
f(p) is prime for all four odd primes of Z/2,310 (3, 5, 7, 11). Hits the same composite wall at 13 as the Cunningham chain. f(7) = 41 governs Artin's conjecture for 7.
41
The 13th prime. Appears in Artin's conjecture as an exceptional index.
f(7) = 41 = forbidden index for 7 as primitive root. f(7) + 1 = 42 = 2*3*7.
Z/12,612,600 vs Z/210
Different rings sharing the same primes {2,3,5,7,11,13}.
Indistinguishable through f(5) and f(11). Distinguished only by f(7) = 41. The polynomial at 7 is the unique discriminator.
CRT roots of f(p)
Root distribution governed by Legendre symbols. Standard quadratic reciprocity.
Among 7 CRT channels, only mod 11 has roots (4 and 8). Legendre (5/p) = +1 only at p = 11. The golden ratio lives in exactly one channel.