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Question 1: Largest Unique number.

Given an integer list [li], define a function that returns the largest integer that only appears once in this list. If there is no such number, return ‘None’ (1 point)

Input: li = [1,1,2,3,5,5,8,9,9,10,10,11,11]

Output: 8

Input: li = [7,7,8,8,9,9,10,10]

Output: ‘None’

Question 2: Stair Climbing

One person is climbing up n stairs. This person can only climb up 1 stair, or 2 stairs at each step. Define a function to show how many different ways this person can choose to climb up to the top. Then, calculate how many ways this person can have for 14 stairs. (2 points)

Input: n=4

Output: 5

Explanation: this person can choose:

[1,1,1,1]; [1,2,1]; [1,1,2]; [2,1,1]; [2,2];

5 ways in total to climb up the stairs.

Input n=5

Output: 8

Explanation: this person can choose:

[1,1,1,1,1]; [1,1,2,1]; [1,1,1,2]; [1,2,1,1]; [1,2,2]; [2,1,2]; [2,2,1];

8 ways in total to climb up the stairs.

Question 3: Goldbach conjecture.

The Goldbach conjecture claims that every even integer greater than two is the sum of two prime numbers. This is an unproved conjecture, however, below 1000000, we can always find prime numbers that fit the condition.

Given an even number n that is greater than 4, define a function that returns a list of 2 prime numbers that sum up to n. (4 points in total)

Input: n = 998

Output: [7,991]

Input: n = 9996

Output: [23, 9973]

Input: n = 100000

Output: [11, 99989]

Hint: you may need multiple functions to finish the final work. Here are 2 functions that may help you:

1, define a function that determine whether a number is a prime number. (1 point for this part)

2, define a function that can find all prime numbers below number n. (1 point for this part)

Submission requirement: First, download the Jupyter notebook from Blackboard, and finish each part with your own codes. You should submit a Jupyter Notebook which contains all codes, inputs, and outputs within it, OR you can submit the PDF file printed by Jupyter Notebook. All test cases must be run, e.g., there are 2 test cases for question 1 and 2, and both outputs must show in your submissions; there are 3 test cases for question 3, and all of them also must be run.