Bernoulli Numbers

The Bernoulli numbers Bₙ appear in the Euler-Maclaurin formula, power sums, and the Riemann zeta function: ζ(2n) = (−1)ⁿ⁺¹ (2π)²ⁿ B₂ₙ / (2 (2n)!). They encode deep arithmetic — the numerators of B₂ₙ/2n detect irregular primes (Kummer's criterion for Fermat's Last Theorem).

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Bernoulli Numbers

nBₙ (exact)Decimal
ζ(2) = π²/6
ζ(4) = π⁴/90
ζ(2n) = (−1)ⁿ⁺¹ (2π)²ⁿ B₂ₙ / 2·(2n)!
B₀=1, B₁=−½, B₂=1/6…

Generating function:
t/(eᵗ−1) = Σ Bₙ tⁿ/n!