Question

Define a procedure unique-pairs that, given an integer \(n\), generates the sequence of pairs \((i, j)\) with \(1 \leq j

Short answer

Use nested loops to generate pairs \((i, j)\), then filter pairs by primality of \(i + j\).

Step-by-step solution

Understanding the Problem

We need to create a function `unique-pairs` that will generate pairs of integers \((i, j)\) such that \(1 \leq j < i \leq n\). The function should output a list of all such pairs for a given integer \(n\). Then, we should use this function to simplify the definition of `prime-sum-pairs`.

Define the unique-pairs Procedure

First, we write the `unique-pairs` procedure. To generate the sequence \((i, j)\), we need to iterate \(i\) from 2 to \(n\) and for each \(i\), iterate \(j\) from 1 to \(i-1\). The pseudocode would look something like this:```pythondef unique_pairs(n): pairs = [] for i in range(2, n + 1): for j in range(1, i): pairs.append((i, j)) return pairs```

Implement the unique-pairs Procedure

Implement the procedure from the pseudocode into actual code. Use nested loops where the outer loop runs from 2 to \(n\) (inclusive), and the inner loop runs from 1 up to \(i-1\). Each time through the inner loop, add the pair \((i, j)\) to the list of pairs.

Test unique-pairs Procedure

To ensure the procedure works, test it with a small value of \(n\), such as 4. The result should be the list: \([(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)]\). Check if the function outputs these pairs correctly.

Connect unique-pairs to prime-sum-pairs

Now, use the `unique-pairs` function to redefine or simplify another function called `prime-sum-pairs`. This function will use `unique-pairs` to filter and output pairs \((i, j)\) from `unique-pairs(n)` where \(i + j\) is a prime number.

Implement prime-sum-pairs Using unique-pairs

Define `prime-sum-pairs` by using the `unique-pairs` output and checking for the primality of the sum of each pair \((i, j)\). Use a helper function to check if a number is prime. For each pair, add it to the results if \(i + j\) is prime.

Key concepts

pair generation

Pair generation is a fundamental concept in algorithm design where the goal is to create every possible combination of pairs from a given set of integers. In the context of the exercise, the task is to generate pairs \(i, j\) such that \(1 \leq j < i \leq n\). This means we want every pair where the second number is always smaller than the first one.
The process begins by deciding the range of integers we will be working with, which is controlled by the integer \(n\). The pairs are formed by looping through potential values for \(i\) and \(j\). Each time we find a valid combination, we create a new pair. This technique is useful in many applications, such as generating combinations for probability, simplifying calculations, or exploring mathematical sequences.

loop iteration

Loop iteration is a technique used when you want to repeat a set of instructions a certain number of times. In our exercise, we use nested loops to systematically generate every valid pair.
The outer loop controls the value of \(i\), starting from 2 up to \(n\), as the first element of our pair \(i\) has to be greater than \(j\). The inner loop iterates through values for \(j\), ranging from 1 to \(i-1\). By using nested loops, we ensure that for every value of \(i\), we consider all smaller numbers \(j\).
Here is a simplified structure:

• Start with \(i = 2\) and increase \(i\) until \(n\)
• For each \(i\), iterate \(j\) from 1 up to \(i-1\)
• Each iteration generates a pair \((i, j)\)

This structured approach ensures that all possible pairs are covered efficiently.

primality testing

Primality testing is the process of determining whether a given number is a prime number. A prime number is defined as a number greater than 1 that cannot be divided evenly by any other numbers except for 1 and itself.
In the exercise, this concept is used to filter pairs generated from the `unique-pairs` procedure by checking if the sum \(i + j\) is prime. To do this, a helper function is typically employed. This function checks if a number \(n\) has divisors other than 1 and \(n\) itself.A straightforward algorithm is to trial divide \(n\) by all integers from 2 to the square root of \(n\) (since a larger factor of \(n\) must be paired with a smaller factor that has already been checked):

• If \(n <= 1\), return False (it's not prime)
• Check all numbers \(k\) from 2 up to \( ext{int}( ext{sqrt}(n))\)
• If \(n\) is divisible by any \(k\), return False
• If no divisors are found, return True

This is an effective way to ensure that a number is prime before using it in further calculations.

function implementation

Function implementation is the practical step where we take our planned outline (the pseudocode) and translate it into working code. This involves writing the syntax for our loops, condition checks, and handling outputs.
For this exercise, we are primarily implementing two functions: `unique_pairs` and `prime_sum_pairs`. The function `unique_pairs(n)` is responsible for looping through integers and generating pairs using nested loops. We then store these pairs in a list and return it.The second function, `prime_sum_pairs`, leverages what `unique_pairs` generates. It takes the list of pairs, then applies a primality test on the sum of each pair.

• Start by calling `unique_pairs` to get a list of pairs
• Iterate through each pair \((i, j)\) and calculate \(i + j\)
• Use a helper function to check if \(i + j\) is a prime number
• If prime, add this pair to the result list

By carefully implementing these functions, we ensure the correct and expected output to the problem. Ensuring each step logically connects and is error-free entails debugging and running tests.