{&/x!:\2+!_x^1%2}
The braces around an expression denote a function. Because this
function contains a single variable, x, it is a monadic
function or a monad. When typing in the above code at the
Klong prompt, the interpreter will respond :monad.
You can apply the function to a value, say 251, using the Apply
(@) verb, e.g.:
{&/x!:\2+!_x^1%2}@251
Expressions evaluate from the right to the left, so the first thing the function does is 1 Divide 2, giving 0.5:
{&/x!:\2+!_x^1%2}@251
The next operation is x Power (1 Divide 2). x1/2 is the square root of x, so this part of the expression computes the square root of 251:
{&/x!:\2+!_x^1%2}@251
Floor (_) then floors (rounds toward −infinity)
that value. So _x^1%2 is Floor (x Power (1 Divide 2)),
in this case the floor of the square root of 251.
{&/x!:\2+!_x^1%2}@251
Enumerate (!) creates a vector containing all integers
from zero up to, but not including, that value. In this case,
!_251^1%2 = !_15.84... =
!15 = [0 ... 14] (a vector of the numbers
from 0 to 14).
{&/x!:\2+!_x^1%2}@251
A number plus a vector gives a new vector with the number added to each
of its elements, so 2+ adds 2 to each element from the
previous step: 2+[0 ... 14] = [2 ... 16].
{&/x!:\2+!_x^1%2}@251
!:\ is a combination of the verb ! (Remainder)
and the adverb :\ (Each-Left). x Remainder-Each-Left
applies Remainder to each element of the previous step (v) with
x as its left operand. It creates a new vector containing
(x!v0),(x!v1),..., i.e.:
(251!2),(251!3),(251!4),(251!5)... =
(1),(2),(3),(1),... = [1 2 3 1 ...].
{&/x!:\2+!_x^1%2}@251
&/ is a combination of Min and Over. Min returns the
minimum of two numbers and Over folds Min over a vector, so Min-Over
returns the minimum of a vector:
{&/x!:\2+!_x^1%2}@251
In this case, the minimum of the vector is 1, because 1 is the smallest remainder obtained by dividing 251 by 2..16. In other words, 251 is prime.
If we had applied the function to a non-prime number, like 253, at least one of the elements of the vector would have divided that number with a remainder of 0, so the result of the function would also be 0.
{&/x!:\2+!_x^1%2}@253
Of course, the prime function is sufficiently complex to bind it
to a name using Define (::):
prime::{&/x!:\2+!_x^1%2}
You can then use the function application syntax to apply it to some values:
prime::{&/x!:\2+!_x^1%2}
prime(251)
prime(253)
Of course, the @ operator would also work:
prime@251
BTW, the prime function only works for values greater than
two. What will it return for 0, 1, and 2? Why?