Square numbers hide odd sums, odd divisors and a mental arithmetic trick
Add up the first few odd numbers and you always land on a perfect square: 1 plus 3 plus 5 plus 7 plus 9 makes 25. Squares are also the only positive whole numbers with an odd count of divisors, and they can end only in the digits 0, 1, 4, 5, 6 or 9.
A square number, or perfect square, is an integer multiplied by itself: 9 is 3 times 3. The name comes from geometry, since a square with side n has area n squared, and n dots can be laid out as a square exactly when n is a square, making squares a kind of figurate number alongside triangular and cube numbers. Squares of real numbers are never negative, and a positive integer with no square divisor other than 1 is called square-free.
The gap between neighbouring squares is always an odd number, 2n minus 1, which is why piling up the first n odd numbers gives n squared. Recursive shortcuts exist too: double the previous square, subtract the one before it and add 2, so twice 25 minus 16 plus 2 yields 36. Every square is also the sum of two consecutive triangular numbers.
Subtracting one from a square always splits it into two factors, which means 3 is the only prime sitting one below a square. More broadly, the difference of two squares equals their sum times their difference, a handy trick for mental arithmetic: 47 times 53 is 50 squared minus 3 squared, or 2500 minus 9, giving 2491. Divisors normally come in pairs, but a square's integer root pairs with itself, leaving an odd total.
Lagrange's four-square theorem says every positive integer is a sum of at most four squares, and some, including numbers of the form 8m + 7, genuinely need all four. A number is a sum of two squares precisely when no prime of the form 4k + 3 appears to an odd power in its factorisation, a line of thought generalised by Waring's problem.
Source: Square number
