# `⍭` — Prime

Keys: `Alt-Backslash`. Ranks: `0` monadic, `0 0` dyadic

`⍭N`: the prime with `N` primes before it. `⍭0` is 2. Integral inputs;
exact integer results, except that the primality test `1⍭` gives
Booleans.

`⍭⁻¹N`: the number of primes below `N`, which is the index of a prime
`N`.

``` bpl
⍭⍳8               ⍝ [2 3 5 7 11 13 17 19]ₓ
⍭⁻¹⍭⍳4            ⍝ [0 1 2 3]ₓ
⍭⁻¹1 2 3 4 5 6     ⍝ [0 0 1 2 2 3]ₓ
```

`K⍭N` selects an operation by the code `K`. The operations are like J’s
`p:`, with different codes. [Factor](factor.qmd) gives factors and
factor tables.

<table>
<thead>
<tr>
<th>K</th>
<th>Result</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>¯2</code></td>
<td>Previous prime, strictly below N</td>
</tr>
<tr>
<td><code>1</code></td>
<td>N is prime</td>
</tr>
<tr>
<td><code>2</code></td>
<td>Next prime, strictly above N</td>
</tr>
<tr>
<td><code>3</code></td>
<td>Euler’s totient</td>
</tr>
</tbody>
</table>

``` bpl
1⍭¯1 0 1 2 3 4     ⍝ $f $f $f $t $t $f
2⍭1 2 3 4 5        ⍝ [2 3 5 5 7]ₓ
¯2⍭3 4 5 6         ⍝ [2 3 3 5]ₓ
3⍭1 2 3 4 5 6      ⍝ [1 1 2 2 4 2]ₓ
```

Euler’s totient of `N` is `N` times the product of `1-÷p` over the
distinct prime factors `p` of `N`. For one exact `N`, `3⍭N` gives the
same result as `N××/1ₓ-÷∪⨸N`. The `ₓ` on `1` keeps the result exact.

``` bpl
N←700ₓ ⋄ N××/1ₓ-÷∪⨸N     ⍝ 240ₓ
3⍭700                    ⍝ 240ₓ
```

Unit cells; results assemble with fill. Primality testing is
deterministic through 64 bits, probabilistic above that (false-positive
bound `2⁻⁶⁴`).

## Errors

- `DOMAIN`: invalid selector/input; previous prime at/below 2