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3 is the smallest odd prime number, and the only prime number that is one more than another prime (namely, 2).
Membership in core sequences
Odd numbers
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1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ...
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A005408
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Prime numbers
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2, 3, 5, 7, 11, 13, 17, 19, ...
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A000040
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Fibonacci numbers
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1, 1, 2, 3, 5, 8, 13, 21, ...
|
A000045
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Lucas numbers
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2, 1, 3, 4, 7, 11, 18, 29, 47, ...
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A000032
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Triangular numbers
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1, 3, 6, 10, 15, 21, 28, 36, 45, ...
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A000217
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In Pascal's triangle, 3 occurs twice. (In Lozanić's triangle, 3 occurs four times).
Sequences pertaining to 3
Multiples of 3
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0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
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A008585
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Inert rational primes in
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5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, ...
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A003630
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Positive numbers of the form
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1, 4, 6, 9, 13, 16, 22, 24, 25, 33, 36, 37, 46, ...
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A084916
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Negative numbers of the form
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–2, –3, –8, –11, –12, –18, –23, –26, –27, –32, ...
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A084917
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sequence beginning at 3
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3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...
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A033478
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Fermat pseudoprimes to base 3
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91, 121, 286, 671, 703, 949, 1105, 1541, 1729, ...
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A005935
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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.
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1.73205080
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A002194
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3 2
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9
|
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1.44224957
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A002581
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3 3
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27
|
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1.31607401
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A011002
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3 4
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81
|
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1.24573093
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A005532
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3 5
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243
|
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1.20093695
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A246708
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3 6
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729
|
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1.16993081
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A246709
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3 7
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2187
|
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1.14720269
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A246710
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3 8
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6561
|
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1.12983096
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A011446
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3 9
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19683
|
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1.11612317
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A246711
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3 10
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59049
|
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1.10503150
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3 11
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177147
|
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1.09587269
|
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3 12
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531441
|
|
|
|
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A000244
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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.
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0.63092975
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A102525
|
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1.58496250
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A102525
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2 3
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8
|
|
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0.91023922
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A121935
|
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1.09861228
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A002162
|
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20.08553692
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A091933
|
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1.00000000
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3 3
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27
|
|
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1.04197804
|
|
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0.95971311
|
|
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31.00627668
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A091925
|
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1.26185950
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A100831
|
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0.79248125
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A094148
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4 3
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64
|
|
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1.46497352
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A113209
|
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0.68260619
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A152914
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5 3
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125
|
|
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1.63092975
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A153459
|
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0.61314719
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A152935
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6 3
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216
|
|
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1.77124374
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A152565
|
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0.56457503
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A152945
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7 3
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343
|
|
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1.89278926
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A113210
|
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0.52832083
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A152956
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8 3
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512
|
|
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2.00000000
|
|
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0.50000000
|
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9 3
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729
|
|
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2.09590327
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A152566
|
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0.47712125
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A114490
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10 3
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1000
|
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(See A000578 for integer cubes).
Values for number theoretic functions with 3 as an argument
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–1
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|
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–1
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2
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4
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2
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2
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1
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1
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2
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This is the Carmichael lambda function.
|
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–1
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This is the Liouville lambda function.
|
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1.2020569031595942853997381615... (see A002117)
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3!
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6
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2
|
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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.
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1
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Unit
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2
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Prime
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3
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. Since , this is tantamount to .
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4
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2 2
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5
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Prime
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6
|
|
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7
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. This may seem clearer expressed as .
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Prime
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8
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2 3. Note that is not a distinct factorization. See 8 for a full explanation.
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9
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10
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2 × 5
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11
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Prime
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12
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|
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13
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|
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14
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|
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15
|
|
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16
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2 4
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17
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Prime
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18
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19
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Prime
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20
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2 2 × 5
|
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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.
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Prime
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Prime
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Irreducible
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Prime
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Prime
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Irreducible
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Prime
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Prime
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Irreducible
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Prime
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Irreducible
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Prime
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Prime
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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