Glossary¶

Terms as Numerisect uses them. Where a word is used loosely elsewhere, the entry says so.

Result strength¶

Proven: A mathematical proof exists. Not a probability statement.
Probable prime: An integer that passed a strong probabilistic test, almost always Baillie–PSW. Not a proof. Numerisect never calls one prime without qualification.
Inconclusive: The question was not settled within the budget or bound applied. Not a negative result.
Exploratory: Indicative rather than certified: a plot sample, an enclosure midpoint, or a value predicted by a conjecture.
Certified enclosure: An interval, produced by ball arithmetic, guaranteed to contain the true value. A proof about a real number.

Fermat pseudoprime to base \(a\): Composite \(n\) with \(a^{n-1} \equiv 1 \pmod n\).
Carmichael number: Composite \(n\) that is a Fermat pseudoprime to every base coprime to \(n\). The smallest is 561. Korselt's criterion characterises them.
Strong pseudoprime: Composite passing the Miller–Rabin test for a given base. The smallest strong pseudoprime to bases 2, 3, 5 and 7 is 3215031751.
Strong liar: A base to which a composite is a strong pseudoprime. At most a quarter of bases are strong liars for any odd composite above 9.
BPSW: Baillie–PSW: a base-2 strong test combined with a Lucas test using Selfridge's parameters. No counterexample is known; none is proven not to exist.
Witness set: A fixed set of bases sufficient to decide primality deterministically below a proven bound.
Primality certificate: A structure allowing independent verification that a number is prime, typically a Pratt tree or an ECPP certificate.

Smooth: An integer whose prime factors are all below a bound \(B\). Powersmooth requires the prime powers to be below \(B\).
Cofactor: What remains after dividing out known factors. A composite cofactor is not a completed factorization.
Pretest: ECM run before a heavier method, to remove factors small enough that the heavier method would waste effort on them. Measured in digits: "t20" means work sufficient to find most 20-digit factors.
Ambiguous form: A binary quadratic form equal to its own inverse in the class group. Finding one during SQUFOF yields a factor.
Special form: An input with algebraic structure, such as \(a^n \pm 1\) or a cyclotomic value, which SNFS can exploit for far less work than general NFS.
Aurifeuillean factorization: An algebraic splitting of \(a^n \pm 1\) beyond the obvious cyclotomic factors.

\(\pi(x)\): The number of primes not exceeding \(x\).
\(\mathrm{li}(x)\): The logarithmic integral, a much better approximation to \(\pi(x)\) than \(x/\log x\).
Merit: A prime gap divided by \(\log p\), measuring how large a gap is relative to the average spacing there.
Maximal gap: A gap larger than every gap before it.
Admissible tuple: A pattern of offsets that does not cover every residue class modulo any prime, hence not obstructed from occurring infinitely often.
Singular series: The constant in the Hardy–Littlewood conjectured density for a tuple pattern.

Critical line: \(\mathrm{Re}(s) = 1/2\), where the Riemann hypothesis asserts all nontrivial zeros lie.
Hardy Z function: A real-valued function on the critical line with the same zeros as \(\zeta\), which is why sign changes can locate them.
Gram point: A point where the Riemann–Siegel theta function is a multiple of \(\pi\). Gram's law is a tendency, not a theorem, and its exceptions are catalogued.
Turing method: A rigorous technique for confirming a zero count through a given height.
\(S(T)\): The remainder in the zero-counting formula.

Tagged output: The TAG:value line protocol Numerisect uses to read engine results, ending in a mandatory completion marker.
Completion marker: The final line an engine prints. Its absence is an error, never an empty success.
Work unit: A parcel of sieving work CADO-NFS hands to a client.
Presieve: Removing multiples of small primes before applying an expensive test to the survivors.