Lisp - prime number

Question 1

you have few missteps there:

(defun is-prime (n d) 
  (if (= d 1) 
      (print "t") 
      (if (= (% n d) 0) 
          (print "nil")

First of all, don't print your results, just return them. Second, there's no % function, it's rem.

The real error is how you make the recursive call. You have an extra pair of parentheses there:

          (is-prime (n (- d 1) )))))
          ;         ^          ^
          ;        this      and this

in Lisp, parentheses signify a function call; but you don't intend to call n with an argument (- d 1), they both are arguments to is-prime. So we just need to remove those extra parentheses,

          (is-prime  n (- d 1)  ))))

So what does it do? It counts down: d, (- d 1) ... 1. And when (= d 1), it returns t. So, one way to call it is

(defun is-prime (n &optional (d (- n 1))) 
  (or (= d 1)
      (and (/= (rem n d) 0)
           (is-prime  n (- d 1)))))

but it is not the most efficient way, :) nor the most safe one, either.

It is much better to count up, not down, for one thing, because any random number is far more likely to have a smaller factor than a larger one. Then, it lets us optimize where we stop -- and stopping at the sqrt is much much more efficient, and just as correct.

Question 2

Well, you're halfway there.

Here's my explanation in English:

You have written in lisp a function is-prime (btw, "yes or no" functions like that are usually named whatever-p in lisp) that tells you if n is relatively prime to d.

What you need to do is go through all d's less than n, and if it's not relatively prime to any of them, return nil, but if after that loop you haven't returned nil, then return t. Your friend here is the function "mod", which tells you whether there is a remainder when its first argument is divided by its second.

Something like:

(defun primep (n)
 (cond
  ((= 1    n)          nil)
   (t
   (loop
     :with root     = (isqrt n)
     :with divisors = (loop :for i :from 3 :to root :by 2 :collect i)
     :for d = (pop divisors)
     :if (zerop (mod n d))
     :do (return nil)
     :else :do (setf divisors (delete-if (lambda (x) (zerop (mod x d))) divisors))
     :while divisors
     :finally (return t)))))

Also, don't print nil, just return nil.

Question 3

I have taken Will Ness's answer as an inspiration and modified the function. Here is the code. Is checks whether 1 is passed as a parameter and makes sure to output nil should that be the case.

(defun is-prime (n &optional (d (- n 1))) 
  (if (/= n 1) (or (= d 1)
      (and (/= (rem n d) 0)
           (is-prime  n (- d 1)))) ()))

I am still learning Common Lisp, so if there are problems with this, please let me know.