Squares

Square numbers

A number is a perfect square if it is the result of multiplying an integer (n) by itself. (n × n or n2)

The sign of a square number

A square number is always positive.

It is the result of multiplying two positive numbers together or two negative numbers, which always gives a positive result.

The last digit of a square number

The last digit of a square number must be 0, 1, 4, 6 or 9

  • 0 × 0 = 0
  • 1 × 1 = 1
  • 2 × 2 = 4
  • 3 × 3 = 9
  • 4 × 4 = 16
  • 5 × 5 = 25
  • 6 × 6 = 36
  • 7 × 7 = 49
  • 8 × 8 = 64
  • 9 × 9 = 81
The difference of two squares

The difference between a2 and b2 is equal to the sum of a and b times the difference between a and b

  • a2 - b2 = (a - b)(a + b)
Expression for the sum of the first n square numbers

n

r2

Σ

r = 1

=
n(n+1)(2n+1)
6

Square numbers

A number is a perfect square if it is the result of multiplying an integer (n) by itself. (n × n or n2)

List of square numbers

Enter the upper range for square values:
1 4 9 16 25 36 49 64 81 100

Relationship between odd numbers and square numbers

The sum of the first n odd numbers equals the nth square number.

r

2r + 1

Σ

n = 1

=

n(n+1) - n = n2
1 2 5 10 17 26 37 50 65 82 1 = 1
4 3 6 11 18 27 38 51 66 83 4 = 1 + 3
9 8 7 12 19 28 39 52 67 84 9 = 1 + 3 + 5
16 15 14 13 20 29 40 53 68 85 16 = 1 + 3 + 5 + 7
25 24 23 22 21 30 41 54 69 86 25 = 1 + 3 + 5 + 7 + 9
36 35 34 33 32 31 42 55 70 87 36 = 1 + 3 + 5 + 7 + 9 + 11
49 48 47 46 45 44 43 56 71 88 49 = 1 + 3 + 5 + 7 + 9 + 11 + 13
64 63 62 61 60 59 58 57 72 89 64 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15
81 80 79 78 77 76 75 74 73 90 81 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17
100 99 98 97 96 95 94 93 92 91 100 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

The product of four consecutive positive integers

The product of four consecutive positive integers plus one is a square number

n(n + 1)(n + 2)(n + 3) + 1 = x2 where x is an integer.

Examples

1 × 2 × 3 × 4 + 1 = 52

2 × 3 × 4 × 5 + 1 = 112

3 × 4 × 5 × 6 + 1 = 192

4 × 5 × 6 × 7 + 1 = 292

5 × 6 × 7 × 8 + 1 = 412

6 × 7 × 8 × 9 + 1 = 552

7 × 8 × 9 × 10 + 1 = 712

8 × 9 × 10 × 11 + 1 = 892

9 × 10 × 11 × 12 + 1 = 1092

10 × 11 × 12 × 13 + 1 = 1312

Proof

n(n + 1)(n + 2)(n + 3) + 1 = n4 + 6n3 + 11n2 + 6n + 1

= (n2 + 3n + 1)2

Lagrange's four-square theorem

Every positive integer can be represented as the sum of four squares

Enter a value:
4 is a square

4 = 22

4 is a sum of four squares

4 = 12 + 12 + 12+ 12

Lagrange's Theorem: 4 can be represented as the sum of four squares

4 = 12 + 12 + 12+ 12