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Nonsquares
From OeisWiki
In a given ring , the nonsquares are numbers that can't be expressed in the form for some . In , the nonsquares are the numbers that are not perfect squares (see A000037 for the positive nonsquares). All squarefree numbers are nonsquares, but not all nonsquares are squarefree (see A051144).
Given an integer , the squares of are still squares in , but some numbers that are nonsquares in are squares in . For example, 2 is a nonsquare in , but a square in , since .
3 | 2 | |
4 | 2, 3 | |
5 | 2, 3 | |
6 | 2, 5 | |
7 | 3, 5 | |
8 | 2, 3, 5, 6, 7 | |
9 | 2, 3, 5, 6, 8 | |
10 | 2, 3, 7, 8 | |
11 | 2, 6, 7, 8, 10 | |
12 | 2, 3, 5, 6, 7, 8, 10, 11 | |
13 | 2, 5, 6, 7, 8, 11 | A028726 |
14 | 3, 5, 6, 10, 12, 13 | A028727 |
15 | 2, 3, 5, 7, 8, 11, 12, 13, 14 | A028728 |
16 | 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 | A028729 |
17 | 3, 5, 6, 7, 10, 11, 12, 14 | A028730 |
18 | 2, 3, 5, 6, 8, 11, 12, 14, 15, 17 | A028731 |
19 | 2, 3, 8, 10, 12, 13, 14, 15, 18 | A028732 |
20 | 2, 3, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19 | A028733 |
21 | 2, 3, 5, 6, 8, 10, 11, 12, 13, 14, 17, 19, 20 | A028734 |
22 | 2, 6, 7, 8, 10, 11, 13, 17, 18, 19, 21 | A028735 |
23 | 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 | A028736 |
24 | 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23 | A028737 |
25 | 2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 23 | A028738 |
26 | 2, 5, 6, 7, 8, 11, 15, 18, 19, 20, 21, 24 | A028739 |
27 | 2, 3, 5, 6, 8, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26 | A028740 |
28 | 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 26, 27 | A028741 |
29 | ||
30 | ||
31 | ||
32 | ||
33 | ||
34 | ||
35 | ||
36 | ||
37 | ||
38 | ||
39 | ||
40 | ||
41 | ||
42 | ||
255 | ||
65537 | 3, 5, 6, 7, 10, 11, 12, 14, 20, 22, 23, 24, 27, 28, 29, 31, 39, 40, 41, 43, 44, 45, 46, 47, ... | A136806 |
If 's squarefree and 's an odd prime in , and coprime to , 's also a prime in if 's a nonsquare in . For example, 3's prime in as is insoluble in integers. However, 5's not prime since , and consequently .