Bernoulli Number Computations: B_8, B_10, B_12

Source: Grok (xAI) analysis, 2026-03-29

Explicit step-by-step computations using the standard recurrence relation for Bernoulli numbers.

Recurrence Formula

For n >= 1:
B_n = -(1/(n+1)) sum(C(n+1, k) B_k, k=0 to n-1)

With B_0 = 1, B_1 = -1/2, and B_m = 0 for odd m > 1.

Known Values

| n | B_n | Decimal |
|---|-----|---------|
| 0 | 1 | 1.0 |
| 1 | -1/2 | -0.5 |
| 2 | 1/6 | 0.1667 |
| 4 | -1/30 | -0.0333 |
| 6 | 1/42 | 0.0238 |

B_8 Computation

B_8 = -(1/9) sum(C(9,k)B_k, k=0 to 7)

Using known values (odd k>1 contribute 0):
Sum = C(9,0)1 + C(9,1)(-1/2) + C(9,2)(1/6) + C(9,4)(-1/30) + C(9,6)*(1/42)
= 1 - 9/2 + 36/6 + 126(-1/30) + 84(1/42)
= 1 - 4.5 + 6 - 4.2 + 2 = 3/10

B_8 = -(1/9) * (3/10) = -3/90 = -1/30

B_10 Computation

B_10 = -(1/11) sum(C(11,k)B_k, k=0 to 9)

Non-zero terms:

k=0: C(11,0)*1 = 1

k=1: C(11,1)*(-1/2) = -11/2

k=2: C(11,2)*(1/6) = 55/6

k=4: C(11,4)(-1/30) = 330(-1/30) = -11

k=6: C(11,6)(1/42) = 462(1/42) = 11

k=8: C(11,8)(-1/30) = 165(-1/30) = -11/2

Sum with common denominator 6:
= (6 - 33 + 55 - 66 + 66 - 33)/6 = -5/6

B_10 = -(1/11) * (-5/6) = 5/66 = 5/66

B_12 Computation

B_12 = -(1/13) sum(C(13,k)B_k, k=0 to 11)

After evaluating all binomial coefficients and weighted sum:
Inner sum = 691/210

B_12 = -(1/13) * (691/210) = -691/2730 = -691/2730

Note: 691 is prime and does not divide 2730, so fraction is in lowest terms.

Complete Table Through B_12

| n | B_n | Decimal |
|---|-----|---------|
| 0 | 1 | 1.0 |
| 1 | -1/2 | -0.5 |
| 2 | 1/6 | 0.16667 |
| 4 | -1/30 | -0.03333 |
| 6 | 1/42 | 0.02381 |
| 8 | -1/30 | -0.03333 |
| 10 | 5/66 | 0.07576 |
| 12 | -691/2730 | -0.25311 |

Even indices alternate sign. Magnitudes grow rapidly for larger indices.

Stirling Series Contributions

Each B_{2k} generates a correction term in ln(n!):

| B_n | Stirling term | Coefficient |
|-----|--------------|-------------|
| B_2 = 1/6 | +1/(12n) | 1/12 |
| B_4 = -1/30 | -1/(360n^3) | -1/360 |
| B_6 = 1/42 | +1/(1260n^5) | 1/1260 |
| B_8 = -1/30 | -1/(1680n^7) | -1/1680 |
| B_10 = 5/66 | +1/(1188n^9) | 5/5940 |
| B_12 = -691/2730 | -691/(360360n^11) | -691/360360 |

Amundson Framework Relevance

These Bernoulli numbers support:

High-precision verification of G(n) convergence toward 1/e

Controlled asymptotic corrections for exact G(n) rationals vs continuous limits

Discretization gap kappa computation

Efficient exact-fraction computation on edge hardware (Pi + Hailo fleet)

The recurrence generates values sequentially without floating-point drift