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
1
4 9 16 25 36 49 64 81 100
Last digit
The last digit of a square number must be 0, 1, 4, 6 or 9
Sum of odd numbers
n2 is the sum of the first n odd numbers.
| 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 |
List of Pythagorean triplets: a2 = b2 + c2
52 = 32 + 42
102 = 62 + 82
132 = 52 + 122
152 = 92 + 122
172 = 82 + 152
202 = 122 + 162
252 = 152 + 202
252 = 72 + 242
262 = 102 + 242
292 = 202 + 212
302 = 182 + 242
342 = 162 + 302
352 = 212 + 282
372 = 122 + 352
392 = 152 + 362
402 = 242 + 322
412 = 92 + 402
452 = 272 + 362
502 = 302 + 402
502 = 142 + 482
512 = 242 + 452
522 = 202 + 482
532 = 282 + 452
552 = 332 + 442
582 = 402 + 422
602 = 362 + 482
612 = 112 + 602
652 = 392 + 522
652 = 332 + 562
652 = 252 + 602
652 = 162 + 632
682 = 322 + 602
702 = 422 + 562
732 = 482 + 552
742 = 242 + 702
752 = 452 + 602
752 = 212 + 722
782 = 302 + 722
802 = 482 + 642
822 = 182 + 802
852 = 512 + 682
852 = 402 + 752
852 = 362 + 772
852 = 132 + 842
872 = 602 + 632
892 = 392 + 802
902 = 542 + 722
912 = 352 + 842
952 = 572 + 762
972 = 652 + 722
Triplets where a is prime
52 = 32 + 42
132 = 52 + 122
172 = 82 + 152
292 = 202 + 212
372 = 122 + 352
412 = 92 + 402
532 = 282 + 452
612 = 112 + 602
732 = 482 + 552
892 = 392 + 802
972 = 652 + 722
Triplets with a = (n2+1)÷2, b = n and c = (n2-1)÷2
52 = 32 + 42
132 = 52 + 122
252 = 72 + 242
412 = 92 + 402
612 = 112 + 602
852 = 132 + 842
Triplets with a = n2+1, b = 2n and c = n2+1
102 = 62 + 82
172 = 82 + 152
262 = 102 + 242
372 = 122 + 352
502 = 142 + 482
652 = 162 + 632
822 = 182 + 802
No paired triplets or triplets with a GCF greater than 1
292 = 202 + 212
532 = 282 + 452
652 = 332 + 562
732 = 482 + 552
852 = 362 + 772
892 = 392 + 802
972 = 652 + 722
Lagrange's four-square theorem
Every positive integer can be represented as the sum of four squares
Square number properties
A number is a perfect square if it is the result of multiplying an integer (n) by itself. (n × n or n2)
Difference of two squares
a2 - b2 = (a - b)(a + b)
Sum of the first n square numbers
Sum of the first n odd numbers is the nth number squared
The product of four consquetive positive integers is square
n(n+1)(n+2)(n+3) + 1 = (n2 + 3n + 1)2
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
