Integer as Sums and Differences of Consecutive Squares/Examples

Example of use of Integer as Sums and Differences of Consecutive Squares

Since the proof above is constructive, we can follow the proof and derive:

\(\ds 15\)

\(=\)

\(\ds 3 + 4 + 4 + 4\)

\(\ds \)

\(=\)

\(\ds \paren {- 1^2 + 2^2} + \paren {3^2 - 4^2 - 5^2 + 6^2} + \paren {7^2 - 8^2 - 9^2 + 10^2} + \paren {11^2 - 12^2 - 13^2 + 14^2}\)

and its associated family of solutions:

\(\ds 15\)

\(=\)

\(\ds - 1^2 + 2^2 + 3^2 - 4^2 - 5^2 + 6^2 + 7^2 - 8^2 - 9^2 + 10^2 + 11^2 - 12^2 - 13^2 + 14^2 + 15^2 - 16^2 - 17^2 + 18^2 - 19^2 + 20^2 + 21^2 - 22^2\)

and so on.

However this may not give the least number of squares that this works for.

In fact we have:

\(\ds 15\)

\(=\)

\(\ds 1^2 - 2^2 + 3^2 - 4^2 + 5^2\)

from Triangular Number as Alternating Sum and Difference of Squares.