Answers to LaGrange's Four Square Theorem Activity
More than one representation is often possible. Only one is given.
1 = 1
2 = 1 + 1
3 = 1 + 1 + 1
4 = 4
5 = 4 + 1
6 = 4 + 1 + 1
7 = 4 + 1 + 1 + 1
8 = 4 + 4
9 = 9
10 = 9 + 1
11 = 9 + 1 + 1
12 = 9 + 1 + 1 + 1
13 = 9 + 4
14 = 9 + 4 + 1
15 = 9 + 4 + 1 + 1
16 = 16
17 = 16 + 1
18 = 16 + 1 + 1
19 = 16 + 1 + 1 + 1
20 = 16 + 4
21 = 16 + 4 + 1
22 = 16 + 4 + 1 + 1
23 = 9 + 9 + 4 + 1
24 = 16 + 4 + 4
25 = 25
26 = 25 + 1
27 = 25 + 1 + 1
28 = 25 + 1 + 1 + 1
29 = 25 + 4
30 = 25 + 4 + 1
48 = 36 + 4 + 4 + 4
56 = 36 + 16 + 4
77 = 64 + 9 + 4
87 = 81 + 4 + 1 + 1
92 = 81 + 9 + 1 + 1
114 = 100 + 9 + 4 + 1
1. Which of the integers from 10 to 20 have more than one representation when expressed as the sum of four or fewer
squares?
Of the integers from 1 to 20, all but 11, 14, and 15 have more than one representation.
2. Attempt to write each of the integers from 1 to 20 as a sum of three or fewer squares. For which of these numbers
can this not be done?
Of the integers from 1 to 20, only 7 and 15 cannot be expressed as the sum of three or fewer squares.
3. Express the year you were born as a sum of four or fewer squares.
1946 = 1936 + 9 + 1
1946 = 1849 + 81 + 16