Question

describe an algorithm, with proof, to compute all possible ways to write a given positive integer n as the sum of squares of consecutive positive integers.

Answer 1

To compute all possible ways to write a given positive integer n as the sum of squares of consecutive positive integers, we can use an algorithm that iterates through the numbers from 1 to n/2 and checks for valid combinations.

Step-by-step explanation:

To compute all possible ways to write a given positive integer n as the sum of squares of consecutive positive integers, we can use the following algorithm:

1. Initialize a variable 'start' to 1.
2. Initialize a variable 'end' to 1.
3. Initialize a variable 'sum' to 1.
4. Initialize an empty list 'result' to store the possible ways.
5. Iterate while 'start' is less than or equal to n/2:
   1. If 'sum' is equal to n, add the range between 'start' and 'end' (inclusive) to 'result'.
   2. If 'sum' is less than n, increment 'end' and add 'end' squared to 'sum'.
   3. If 'sum' is greater than n, subtract 'start' squared from 'sum' and increment 'start'.

The algorithm ensures that we consider all possible consecutive squares starting from 1 and finds the combinations that sum up to n. The proof for this algorithm lies in the property of consecutive squares: (n+1)^2 - n^2 = 2n+1, which ensures that the algorithm covers all possible combinations.