Glossary & Sequence Reference

The n-th prime number, in the natural order: p₁ = 2, p₂ = 3, p₃ = 5, …

OEIS: A000040

The distance from one prime to the next. Always even for n ≥ 2 (since all primes after 2 are odd). The first few are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, …

OEIS: A001223

How the gap is changing — positive when the gap is widening, negative when it is shrinking. Sums of three consecutive primes determine local "curvature" in the prime terrain.

OEIS: A036263

The second difference normalized by the surrounding span. Always lies in [−1, 1] — proof: |Δ²_n| = |g_n − g_n−1| ≤ g_n + g_n−1 = p_n+1 − p_n−1.

r_n = +1 when the right gap is much larger than the left; r_n = −1 when the left is much larger; r_n = 0 when both gaps are equal.

Returned as a reduced rational (num/den with gcd = 1) so the sequence is exact for arbitrary precision. No OEIS entry exists for this normalized form; it appears to be original to this project, suitable for citation as "second ratio" of the prime sequence.

The 1-indexed position of a prime in the natural ordering. Index 1 is p = 2, index 2 is p = 3, etc. Inputs to this site can be either the index or the value — a "by index" search of n=100 gives the 100th prime (541), while a "by value" search of n=100 gives the next prime ≥ 100 (which is 101).

How impressive a gap is, normalized against the "expected" gap at that scale. The Prime Number Theorem implies the average gap near p is roughly ln(p), so the average merit converges to 1 as n grows.

A merit of 4 means the gap is 4× the local average — a noteworthy outlier. Gap-hunters in number theory search for "high-merit gaps". The current record (as of 2024) is around 41.94 by Helm, Andersen et al.

Tests Cramér's heuristic, which predicts that prime gaps satisfy g_n = O(ln²(p_n)). The conjecture is limsup C_n = 1; current empirical evidence puts it at about 0.92.

Where merit asks "how big relative to the average," Cramér asks "how big relative to the conjectured maximum."

For a fixed modulus m, count the primes in each residue class p ≡ a (mod m), excluding the primes that divide m. As n grows, the counts in each admissible class converge (Dirichlet's theorem), but at any finite x the leader fluctuates.

The empirical observation, due to Chebyshev (1853), that primes ≡ 3 (mod 4) lead primes ≡ 1 (mod 4) at "almost all" finite x. The first sign change — where π(x; 4, 1) overtakes π(x; 4, 3) — occurs at x ≈ 26 861 (Bays & Hudson, 1978). The bias generalizes: among admissible residues, those that are quadratic non-residues tend to lead.

OEIS: A007350 (sign changes of π(x;4,3) − π(x;4,1))

The number of times the leader of a residue race switches within the queried window. Littlewood (1914) proved there are infinitely many.

The sum of natural logarithms of all primes up to x. The Prime Number Theorem is equivalent to θ(x) ~ x.

More tractable than π(x) for analytic work. Under the Riemann Hypothesis, the deviation θ(x) − x is bounded by O(√x · ln²(x)). On this site, set n=1, n_type=index to start the sum at p = 2 — the only configuration where the returned total equals true θ(p_last).

OEIS: A083409 (deviation θ(x) − x at integer x)

The empirically-bounded magnitude when the neighborhood actually starts at p = 2. Under RH, this stays bounded by O(ln²(x)), so values larger than that would be a Riemann-Hypothesis-disproving anomaly.

A run of primes matching a fixed gap signature. Examples: twin primes have signature [2] (gap 2), cousin primes [4], sexy primes [6], prime quadruplets [2, 4, 2].

Searched here as ?signature=2,4,2&start=…&limit=… on /api/v1/constellations.

The list of gaps between consecutive primes in the constellation. A signature of length L matches L+1 primes. The offsets [0, g₁, g₁+g₂, …, g₁+…+g_L] are the relative positions of those primes from the first.

A k-tuple is admissible if for every prime q, its offsets do not cover all residue classes mod q. Inadmissible tuples can occur at most finitely often. Example: signature [2, 2] gives offsets {0, 2, 4} which mod 3 is {0, 1, 2} — every class — so no triple of primes can form (p, p+2, p+4) infinitely often (excluding the single triple 3, 5, 7).

Verifiable instantly via /api/v1/constellations/admissible?signature=….

Observed count divided by the natural-log span of the queried window. Hardy–Littlewood predicts asymptotic density ∝ Π(C_k) / ln^k(x) for admissible tuples; this site reports the observed density for direct comparison.

π(x) ~ x / ln(x), or equivalently p_n ~ n · ln(n). Implies the average gap near p is ≈ ln(p), and is the basis for both the merit normalization (M ≈ 1 on average) and the θ(x) ~ x expectation.

The conjecture that all non-trivial zeros of the Riemann zeta function ζ(s) lie on Re(s) = ½. Equivalent to a sharp bound on PNT's error: |θ(x) − x| = O(√x · ln²(x)).

g_n = O(ln²(p_n)) — gaps grow no faster than the square of the local logarithm. Stronger than what PNT alone implies. Currently proven only conditionally (under RH, a weaker bound holds).

The primality test used by this site for any value beyond the precomputed sieve. Combines a Miller–Rabin test with base 2 and a strong Lucas test. No counterexamples are known despite testing against many trillions of candidates; widely accepted as practically deterministic for the bit-sizes this site supports.

Before running BPSW on a candidate, we skip any integer that shares a small prime factor with W = 2 · 3 · 5 · 7 · 11 · 13 = 30 030. Of the 30 030 residues only 5 760 are coprime to W, so 80.8 % of composite candidates are eliminated essentially for free — a 5× speedup on nextPrime walks.

Compensated floating-point summation used when accumulating ln(p) for θ(x). Tracks the running rounding error so the sum stays accurate over thousands of terms.

For primes beyond Number.MAX_SAFE_INTEGER (~9 × 10¹⁵), the naïve Math.log(Number(bigint)) loses precision. Our replacement decomposes ln(n) = (digits − 1) · ln(10) + ln(leading 15 digits / 10¹⁴), preserving ~14 significant digits at any magnitude.