Square Numbers: Geometric Arrays, Digit Invariants & Properties
1. The Definition and Geometric Model
A natural number $m$ is called a square number or perfect square if there exists a natural number $n$ such that:
$$m = n^2 = n \times n$$
GEOMETRIC ARRAY OF SQUARES
1^2 = 1 2^2 = 4 3^2 = 9
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2. Properties of Square Numbers
- Ending Digits Invariant: A square number can ONLY end in 0, 1, 4, 5, 6, or 9.
- Numbers ending in 2, 3, 7, or 8 can NEVER be perfect squares.
- Even Zeros Rule: A perfect square always ends in an even number of zeros ($100, 400, 900, 10000$). Numbers ending in 1, 3, or 5 zeros are never squares.
- Sum of First $n$ Odd Numbers:
$$1 + 3 + 5 + \dots + (2n - 1) = n^2$$
- Non-Square Numbers Between Squares:
Between $n^2$ and $(n+1)^2$, there are exactly $2n$ non-square natural numbers.
- Example: Between $5^2 = 25$ and $6^2 = 36$, there are $2 \times 5 = 10$ non-squares ($26, 27, \dots, 35$).
3. Pythagorean Triplets
Three positive integers $a, b, c$ form a Pythagorean Triplet if $a^2 + b^2 = c^2$.
- For any natural number $m > 1$:
$$\mathbf{(2m)^2 + (m^2 - 1)^2 = (m^2 + 1)^2}$$
- The triplet is given by: $\{2m, m^2 - 1, m^2 + 1\}$.