PrimePairs

The lesser prime $p$ of a pair $(p, p+gap)$, both prime -- $gap$ selects the family ($2$=twin, $4$=cousin, $6$=sexy, …), as a lazy indexed family.

PrimePairs(gap)the prime $p$ with $p+gap$ also prime, an indexed collection of open infinitude for every $gap$.

Typed signatures

Type	Package	Applies
(integer<0..>) -> indexed_collection<integer>	combinatorics	always

Domain: Collections

Details

• A lazy indexed collection for each $gap$; whether there are infinitely many prime pairs at ANY fixed gap is open (the twin-prime conjecture generalises to every gap), so $Count(PrimePairs(gap)) = NaN$ for every $gap$, not just $gap=2$.
• $PrimePairs(2)$, $PrimePairs(4)$, $PrimePairs(6)$ are TwinPrimes, CousinPrimes, SexyPrimes verbatim.
• $At(PrimePairs(gap), i)$ unranks the $i$-th match by scanning forward, testing primality of $n$ and $n+gap$ -- $At(PrimePairs(4), 1) = 3$.
• Membership goes through Element: $Element(3, PrimePairs(4))$ is true ($3,7$ both prime), $Element(5, PrimePairs(4))$ is false ($9$ is not).

Enumeration

Examples

1 more kept as data, checked against the oracles

About

Examples	1

See also: Count, At, Element, TwinPrimes, CousinPrimes, SexyPrimes