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# SICP exercise 2.40

## Problem

Define a procedure unique-pairs that, given an integer n, generates the sequence of pairs (i, j) with [itex]1 \le j < i \le n[/itex]. Use unique-pairs to simplify the definition of prime-sum-pairs given in the text.

## Solution

Here are all of the definitions needed for the version of prime-sum-pairs given in the text:

```(define nil (quote ()))

(define (filter predicate sequence)
(cond ((null? sequence) nil)
((predicate (car sequence))
(cons (car sequence)
(filter predicate (cdr sequence))))
(else (filter predicate (cdr sequence)))))

(define (enumerate-interval low high)
(if (> low high)
nil
(cons low (enumerate-interval (+ low 1) high))))

(define (accumulate op initial sequence)
(if (null? sequence)
initial
(op (car sequence)
(accumulate op initial (cdr sequence)))))

(define (flatmap proc seq)
(accumulate append nil (map proc seq)))

(define (square x)
(* x x))

(define (smallest-divisor n)
(find-divisor n 2))

(define (find-divisor n test-divisor)
(cond ((> (square test-divisor) n) n)
((divides? test-divisor n) test-divisor)
(else (find-divisor n (+ test-divisor 1)))))

(define (divides? a b)
(= (remainder b a) 0))

(define (prime? n)
(= n (smallest-divisor n)))

(define (prime-sum? pair)
(prime? (+ (car pair) (cadr pair))))

(define (make-pair-sum pair)

(define (prime-sum-pairs n)
(map make-pair-sum
(filter prime-sum?
(flatmap
(lambda (i)
(map (lambda (j) (list i j))
(enumerate-interval 1 (- i 1))))
(enumerate-interval 1 n)))))```

Here's a definition of unique-pairs that makes list pairs (instead of Scheme primitive pairs):

```(define (unique-pairs n)
(flatmap (lambda (i)
(map (lambda (j) (list i j))
(enumerate-interval 1 (- i 1))))
(enumerate-interval 1 n)))```

Test:

`(unique-pairs 3)`

Output:

```((2 1) (3 1) (3 2))
```

And here's a simpler definition of prime-sum-pairs that uses unique-pairs:

```(define (prime-sum-pairs n)
(map make-pair-sum
(filter prime-sum? (unique-pairs n))))```

Test:

`(prime-sum-pairs 6)`

Output:

```((2 1 3) (3 2 5) (4 1 5) (4 3 7) (5 2 7) (6 1 7) (6 5 11))
```