Square Numbers
Lagrange's Theorem
A famous mathematician called Lagrange proved that every whole number could be written as the sum of 4 or fewer square numbers.
Examples:
19 = 16 + 1 + 1 + 1
35 = 25 + 9 + 1
47 = 36 + 9 + 1 + 1
What would 96 be?
Please help!!
The answer is 8²+4²+4². There's an applet on the internet that will do the decomposition for you for any positive integer: just google "Dario Alpern Lagrange" and click on the link at the bottom of the page to find it. It's very fast indeed.
To find the representation of the number 96 as the sum of four or fewer square numbers, we can follow these steps:
Step 1: Start with the largest perfect square less than or equal to 96, which is 81 (9^2).
Step 2: Subtract this perfect square from 96 to get the remainder, which is 15 (96 - 81 = 15).
Step 3: Repeat steps 1 and 2 with the remainder. The largest perfect square less than or equal to 15 is 9 (3^2). Subtracting this square from 15, we get 6 (15 - 9 = 6).
Step 4: Repeat steps 1 and 2 with the new remainder. The largest perfect square less than or equal to 6 is 4 (2^2). Subtracting this square from 6, we get 2 (6 - 4 = 2).
Step 5: Finally, use the smallest square numbers possible to make up the remainder. Since 2 is not a perfect square, we can represent it as 1 + 1.
Putting it all together, we have:
96 = 81 + 9 + 4 + 1 + 1
Thus, 96 can be written as the sum of 5 square numbers.