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1: 27.22 Software
  • GIMPS. This includes updates of the largest known Mersenne prime.

  • 2: 27.12 Asymptotic Formulas: Primes
    where λ ( α ) depends only on α , and ϕ ( m ) is the Euler totient function (§27.2). … The largest known prime (2018) is the Mersenne prime 2 82 , 589 , 933 1 . …
    3: Errata
  • Paragraph Prime Number Theorem (in §27.12)

    The largest known prime, which is a Mersenne prime, was updated from 2 43 , 112 , 609 1 (2009) to 2 82 , 589 , 933 1 (2018).

  • 4: 27.19 Methods of Computation: Factorization
    Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time. … As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard p 1 , and a 67-digit prime for ecm. … As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …
    5: 3.8 Nonlinear Equations
    (More precisely, p is the largest of the possible set of indices for (3.8.3).) … For multiple zeros the convergence is linear, but if the multiplicity m is known then quadratic convergence can be restored by multiplying the ratio f ( z n ) / f ( z n ) in (3.8.4) by m . …