TwinPrimes

The (lesser) twin primes $3, 5, 11, 17, 29, …$ : primes $p$ with $p+2$ also prime.

TwinPrimesthe lesser prime

p

of a twin pair

(p, p+2)

, an indexed collection of open infinitude.

Domain: Collections

Details

A lazy indexed collection; whether there are infinitely many is the open twin-prime conjecture, so $Count(TwinPrimes) = NaN$ rather than $+\infty$ -- a deliberately different answer from every $+\infty$ family in this library, marking a genuinely open question rather than a known-infinite one. OEIS A001359 (the LESSER member; A001097 lists both members of each pair instead).
$At(TwinPrimes, k)$ unranks by scanning forward, testing primality of $n$ and $n+2$ -- $At(TwinPrimes, 3) = 11$ .
Membership goes through Element: $Element(11, TwinPrimes)$ is true ( $11, 13$ both prime), $Element(13, TwinPrimes)$ is false ( $15$ is not).
$TwinPrimes$ is $PrimePairs(2)$ verbatim.

Enumeration

Basic #

The first 20 lesser twin primes, OEIS A001359

\mathrm{Take}(\mathrm{TwinPrimes}, 20)

Out

\bigl\lbrack3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311\bigr\rbrack

$Count(TwinPrimes)$ is unknown, but the first 20 exist, so a prefix counts exactly

\mathrm{Count}(\mathrm{Take}(\mathrm{TwinPrimes}, 20))

Out

20

Properties #

11∈\mathrm{TwinPrimes}

Out

\mathrm{True}

13 is not -- $13+2=15$ is not prime

13∈\mathrm{TwinPrimes}

Out

\mathrm{False}

About

Examples	4
Crosswalk	OEIS