This polynomial has roots at {1, 2, 3, 5}. Its coefficients are 11, 41, 61, and 30. Evaluate P at integers from 6 to 13: every result factors entirely into primes from {2, 3, 5, 7, 11}. The factorizations connect to E8, the Fano plane, and factorials.
Evaluations at Integer Points
Shadow Evaluation (PROVED)
P(x) = x^4 - 11x^3 + 41x^2 - 61x + 30. Every evaluation at integers 6 through 13 is 11-smooth (factors only into {2,3,5,7,11}).
x
P(x)
Factorization
Connection
0
30
2 * 3 * 5
Coxeter number of E8
6
60
4 * 3 * 5
lambda of Z/2,310
7
240
16 * 3 * 5
|roots(E8)|
8
630
2 * 9 * 5 * 7
9
1344
64 * 3 * 7
8 * |PSL(2,7)|
10
2520
8 * 9 * 5 * 7
lcm(1..9)
11
4320
32 * 27 * 5
12
6930
2 * 9 * 5 * 7 * 11
3 * primorial(11)
13
10560
64 * 3 * 5 * 11
All 5 primes present
Note: 5 is absent from P(9) = 1344. This is the only evaluation in the range where one of the five primes drops out. At x = 9, the factor (9-5) = 4 contributes only powers of 2.
P(7) = 240 = E8 Root Count
The E8 root system has 240 roots. E8 has rank 8 and Coxeter number 30. Roots = rank * Coxeter = 8 * 30 = 240.
P(7) = 240
= 6 * 5 * 4 * 2
The four factors are (7-1)(7-2)(7-3)(7-5). Product = |roots(E8)|.
P(7)/P(0) = 8
= rank(E8)
P(0) = 30 = Coxeter number. P(7) = 30 * 8. The ratio is the rank.
P(17) = 40,320
= 8!
P at 17 is the factorial of 8. Also: 40,320 = 240 * 168 = |roots(E8)| * |PSL(2,7)|.
The Mirror Identity
Mirror Identity (PROVED)
P(-1) = P(11)/P(0) = 4320/30 = 144. Proof: P(-1) = (-2)(-3)(-4)(-6) = 144. P(11) = 10*9*8*6 = 4320. P(0) = 1*2*3*5 = 30. Ratio = 144. The polynomial at -1 equals the ratio of its value at 11 to its value at 0.
CRT Root Anatomy
CRT Root Theorem (PROVED)
P(x) has 5 roots mod 8 and mod 9 (extra root: -1), but exactly 4 roots modulo 25, 49, 11, 13, and 17. Root count sum across all 7 channels = 30 = P(0). Non-root sum = 102. Discriminant = 2304 = 48^2.
Why 5 roots mod 8 and 9? Because P(-1) = 144 is divisible by 8 and 9 but not by 25 or 49. So -1 is an extra root only in the mod-8 and mod-9 channels.
Channel
Modulus
Roots
Non-roots
mod 8
8
5 (incl. -1)
3
mod 9
9
5 (incl. -1)
4
mod 25
25
4
21
mod 49
49
4
45
mod 11
11
4
7
mod 13
13
4
9
mod 17
17
4
13
Root sum
5+5+4+4+4+4+4 = 30
= P(0). The root count across all channels equals the polynomial evaluated at zero.
Non-root sum
3+4+21+45+7+9+13 = 102
= 2*3*17. The complement is governed by 17, the ring-closing prime.
Discriminant
disc(P) = 2304 = 48^2
= 2^8 * 3^2. Only 2 and 3 (the first two chain primes) appear.
The Ratio Staircase
Ratio Staircase (PROVED)
P(p)/P(0) = 2^k * cofactor for each non-root prime p. Cofactors {1, 9, 11, 21} sum to 42. Powers of 2 sum to 8+16+32+64 = 120 = 5!.
p
P(p)/30
Power of 2
Cofactor
7
8
2^3 = 8
1
11
144
2^4 = 16
9
13
352
2^5 = 32
11
17
1344
2^6 = 64
21
Cofactor sum
1 + 9 + 11 + 21 = 42
= 2*3*7. The cofactors at the four non-root primes sum to 42.
Power-of-2 sum
8+16+32+64 = 120
= 5!. The 2-powers in the staircase sum to the factorial of 5.
P(17)/P(7)
40320/240 = 168
= |PSL(2,7)| = |Aut(Fano)|. The ratio of P at 17 to P at 7 is the Fano plane symmetry group.
Explore: P(x) Evaluator
Enter any integer x. Compute P(x) = (x-1)(x-2)(x-3)(x-5). For x from 6 to 13, every result factors into {2, 3, 5, 7, 11} only.