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3
3 is the smallest odd prime number, and the only prime number that is one more than another prime (namely, 2).
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 3
- 3 Partitions of 3
- 4 Roots and powers of 3
- 5 Logarithms and cubes
- 6 Values for number theoretic functions with 3 as an argument
- 7 Factorization of some small integers in a quadratic integer ring adjoining the square roots of −3, 3
- 8 Factorization of 3 in some quadratic integer rings
- 9 Representation of 3 in various bases
- 10 See also
Membership in core sequences
Odd numbers | 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... | A005408 |
Prime numbers | 2, 3, 5, 7, 11, 13, 17, 19, ... | A000040 |
Fibonacci numbers | 1, 1, 2, 3, 5, 8, 13, 21, ... | A000045 |
Lucas numbers | 2, 1, 3, 4, 7, 11, 18, 29, 47, ... | A000032 |
Triangular numbers | 1, 3, 6, 10, 15, 21, 28, 36, 45, ... | A000217 |
In Pascal's triangle, 3 occurs twice. (In Lozanić's triangle, 3 occurs four times).
Sequences pertaining to 3
Multiples of 3 | 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ... | A008585 |
Inert rational primes in | 5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, ... | A003630 |
Positive numbers of the form | 1, 4, 6, 9, 13, 16, 22, 24, 25, 33, 36, 37, 46, ... | A084916 |
Negative numbers of the form | –2, –3, –8, –11, –12, –18, –23, –26, –27, –32, ... | A084917 |
sequence beginning at 3 | 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ... | A033478 |
Fermat pseudoprimes to base 3 | 91, 121, 286, 671, 703, 949, 1105, 1541, 1729, ... | A005935 |
Partitions of 3
There are only three partitions of 3: {1, 1, 1}, {1, 2} and {3}. Thus the only partition of 3 into primes is a trivial partition.
Roots and powers of 3
In the table below, irrational numbers are given truncated to eight decimal places.
1.73205080 | A002194 | 3 2 | 9 | |
1.44224957 | A002581 | 3 3 | 27 | |
1.31607401 | A011002 | 3 4 | 81 | |
1.24573093 | A005532 | 3 5 | 243 | |
1.20093695 | A246708 | 3 6 | 729 | |
1.16993081 | A246709 | 3 7 | 2187 | |
1.14720269 | A246710 | 3 8 | 6561 | |
1.12983096 | A011446 | 3 9 | 19683 | |
1.11612317 | A246711 | 3 10 | 59049 | |
1.10503150 | 3 11 | 177147 | ||
1.09587269 | 3 12 | 531441 | ||
A000244 |
Logarithms and cubes
In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript.
If is not a multiple of 7, then either or is. Hence the formula for the Legendre symbol .
As above, irrational numbers in the following table are truncated to eight decimal places.
0.63092975 | A102525 | 1.58496250 | A102525 | 2 3 | 8 | |||
0.91023922 | A121935 | 1.09861228 | A002162 | 20.08553692 | A091933 | |||
1.00000000 | 3 3 | 27 | ||||||
1.04197804 | 0.95971311 | 31.00627668 | A091925 | |||||
1.26185950 | A100831 | 0.79248125 | A094148 | 4 3 | 64 | |||
1.46497352 | A113209 | 0.68260619 | A152914 | 5 3 | 125 | |||
1.63092975 | A153459 | 0.61314719 | A152935 | 6 3 | 216 | |||
1.77124374 | A152565 | 0.56457503 | A152945 | 7 3 | 343 | |||
1.89278926 | A113210 | 0.52832083 | A152956 | 8 3 | 512 | |||
2.00000000 | 0.50000000 | 9 3 | 729 | |||||
2.09590327 | A152566 | 0.47712125 | A114490 | 10 3 | 1000 |
(See A000578 for integer cubes).
Values for number theoretic functions with 3 as an argument
–1 | ||
–1 | ||
2 | ||
4 | ||
2 | ||
2 | ||
1 | ||
1 | ||
2 | This is the Carmichael lambda function. | |
–1 | This is the Liouville lambda function. | |
1.2020569031595942853997381615... (see A002117) | ||
3! | 6 | |
2 |
Factorization of some small integers in a quadratic integer ring adjoining the square roots of −3, 3
is actually a subdomain because it is not integrally closed. Since , it is not enough to consider algebraic integers of the form (with ); we must also consider algebraic integers of the form (with odd).
Define . This number is of great importance not only because the ring of algebraic integers is , but also because it is a complex cubic root of unity, meaning that . And since its norm is 1, this means that is a unit of , and so are , and . Thus, together with 1 and –1, this means that has six units, more than any other imaginary quadratic integer ring.
That's very little compared to , which has infinitely many units of the form . Both (also called the domain of Eisenstein integers) and are unique factorization domains. The inertial primes in are the primes congruent to 2 modulo 3 (see A003627). If a prime , then it is inertial in (see A003630).
In the table below, some factorizations in will also be expressed using ; this does not constitute a distinct factorization since has unique factorization and . This is done for the sake of clarification, to facilitate comparison to other domains not usually expressed in terms of multiples of a unit that has both a real and an imaginary part.
1 | Unit | |
2 | Prime | |
3 | . Since , this is tantamount to . | |
4 | 2 2 | |
5 | Prime | |
6 | ||
7 | . This may seem clearer expressed as . | Prime |
8 | 2 3. Note that is not a distinct factorization. See 8 for a full explanation. | |
9 | ||
10 | 2 × 5 | |
11 | Prime | |
12 | ||
13 | ||
14 | ||
15 | ||
16 | 2 4 | |
17 | Prime | |
18 | ||
19 | Prime | |
20 | 2 2 × 5 |
Factorization of 3 in some quadratic integer rings
As was mentioned above, 3 is a prime number in . But it is composite in some quadratic integer rings.
Prime | |||
Prime | |||
Irreducible | Prime | ||
Prime | |||
Irreducible | |||
Prime | |||
Prime | |||
Irreducible | Prime | ||
Irreducible | |||
Prime | |||
Prime |
For through , 3 is irreducible and possibly prime. And beyond , without reference to a specific ring, it can only be guaranteed to be irreducible, as there are no more imaginary quadratic rings with unique factorization after that.
Representation of 3 in various bases
In binary, 3 is represented as 11 and in ternary as 10 (as well as in balanced ternary). For all ordinary higher integer bases, 3 is 3, as well as in quater-imaginary base. In negabinary, 3 is 111, since .
In base 10, as well as in any other case in which the base is congruent to 1 modulo 3, there is a simple divisibility test for 3: if the digital root of a given number is 3 or any multiple thereof, then the number is also divisible by 3. In base 10, this means that all integers having a digital root of 3, 6 or 9 are multiples of 3 (see A008585).
See also
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 |
1729 |