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2
2 is the only even prime number, and thus "the oddest prime."
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 2
- 3 Partitions of 2
- 4 Roots and powers of 2
- 5 Logarithms and squares
- 6 Values for number theoretic functions with 2 as an argument
- 7 Factorization of 2 in some quadratic integer rings
- 8 Factorization of some small integers in a quadratic integer ring adjoining the square roots of −2, 2
- 9 Representation of 2 in various bases
- 10 See also
- 11 References
Membership in core sequences
Even numbers | 0, 2, 4, 6, 8, 10, 12, 14,... | A005843 |
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, ... | A000032 |
Catalan numbers | 1, 1, 2, 5, 14, 42, 132, ... | A000108 |
Factorials | 1, 1, 2, 6, 24, 120, 720, ... | A000142 |
Primorials | 1, 2, 6, 30, 210, 2310, ... | A002110 |
In Pascal's triangle, 2 occurs only once, and is in fact the only positive integer to appear only once in the triangle. 1 occurs infinitely often, while all other integers appear at least twice. (In Lozanić's triangle, 2 occurs four times).
Sequences pertaining to 2
has class number 2 | −5, −6, −10, −13, −15, −22, −35, −37, −51, −58, −91, −115, −123, −187, ... | A005847 |
has class number 2 | 10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 85, 87, 91, 95, ... | A029702 |
Partitions of 2
There are only two partitions of 2: {1, 1} and {2}. Thus the only partition of 2 into primes is a trivial partition.
Roots and powers of 2
In the table below, irrational numbers are given truncated to eight decimal places.
1.41421356 | A002193 | 2 2 | 4 | |
1.25992104 | A002580 | 2 3 | 8 | |
1.18920711 | A010767 | 2 4 | 16 | |
1.14869835 | A005531 | 2 5 | 32 | |
1.12246204 | A010768 | 2 6 | 64 | |
1.10408951 | A010769 | 2 7 | 128 | |
1.09050773 | A010770 | 2 8 | 256 | |
1.08005973 | A010771 | 2 9 | 512 | |
1.07177346 | A010772 | 2 10 | 1024 | |
1.06504108 | A010773 | 2 11 | 2048 | |
1.05946309 | A010774 | 2 12 | 4096 | |
1.05476607 | A010775 | 2 13 | 8192 | |
1.05075663 | A010776 | 2 14 | 16384 | |
1.04729412 | A010777 | 2 15 | 32768 | |
1.04427378 | A010778 | 2 16 | 65536 | |
A000079 |
Of course the roots given above are the principal real roots. There are also negative real roots and complex roots.
- , (both real)
- , (the two complex roots are the same except for the sign of the imaginary part)
- , , ,
- , and so on and so forth.
The number 2 figures in the simple continued fraction for its own principal square root:
And, interestingly enough, also figures in a continued fraction involving the square root of 3:
(See A090388).
Logarithms and squares
In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript. In information theory, the binary logarithm is often of interest.
From Fermat's little theorem, we can deduce that if is not a multiple of 3, then is.
If is not a multiple of 5, 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.
1.00000000 | 2 2 | 4 | ||||||
1.44269504 | A007525 | 0.69314718 | A002162 | 7.38905609 | A072334 | |||
1.58496250 | A020857 | 0.63092975 | A102525 | 3 2 | 9 | |||
1.65149612 | A216582 | 0.60551156 | A104288 | 9.86960440 | A002388 | |||
2.00000000 | A000038 | 0.50000000 | A020761 | 4 2 | 16 | |||
2.32192809 | A020858 | 0.43067655 | A152675 | 5 2 | 25 | |||
2.58496250 | A020859 | 0.38685280 | A152683 | 6 2 | 36 | |||
2.80735492 | A020860 | 0.35620718 | A152713 | 7 2 | 49 | |||
3.00000000 | 0.33333333 | A010701 | 8 2 | 64 | ||||
3.16992500 | A020861 | 0.31546487 | A152747 | 9 2 | 81 | |||
3.32192809 | A020862 | 0.30102999 | A007524 | 10 2 | 100 |
(See A000290 for integer squares).
Values for number theoretic functions with 2 as an argument
–1 | ||
0 | ||
1 | ||
3 | ||
2 | ||
1 | ||
1 | ||
1 | ||
1 | This is the Carmichael lambda function. | |
–1 | This is the Liouville lambda function. | |
1.6449340668482264364724... (see A013661). | ||
2! | 2 | |
1 |
Factorization of 2 in some quadratic integer rings
As was mentioned above, 2 is a prime number in . But it is composite in some quadratic integer rings. In fact, in order for 2 to be a prime in which is a unique factorization domain, the congruence must hold. If instead , this means that 2 is the associate of the square of a prime, while means that 2 is the product of two distinct primes.^{[1]}
Prime | |||
Irreducible | Prime | ||
Irreducible | Irreducible | ||
Prime | |||
Irreducible | Prime | ||
Irreducible | |||
Prime |
Because the norm in an imaginary quadratic ring is never negative, 2 is irreducible in almost all imaginary rings, and in fact the only exceptions are shown in the table above. Whether or not it is also prime is a separate issue, but we can generalize this much: if is even and squarefree, then 2 is irreducible but not prime in , for but not .
The situation is more complicated in real rings. If is positive and even, and is a unique factorization, then 2 is composite, and so is , whether that is a prime or not in . Then the fact that but not is not evidence of multiple factorization.
Factorization of some small integers in a quadratic integer ring adjoining the square roots of −2, 2
is a commutative quadratic integer ring with unity, and a unique factorization domain. Its units are of the form (with ). If an odd prime is congruent of 1 or –1 modulo 8, it is composite in , and obviously so is 2 (see A038873).
Expressing the successive units in as , the sequence of is given by A001333, while is given by the Pell numbers, A000129.
is also a unique factorization domain. But in it, there are only two units: 1 and –1. The inertial primes in are the primes in that are congruent to 5 or 7 modulo 8.
1 | Unit | |
2 | ||
3 | Prime | |
4 | ||
5 | Prime | |
6 | ||
7 | Prime | |
8 | ||
9 | 3 2 | |
10 | ||
11 | Prime | |
12 | ||
13 | Prime | |
14 | ||
15 | 3 × 5 | |
16 | ||
17 | ||
18 | ||
19 | Prime | |
20 |
We then say that is norm-Euclidean, and from this it automatically follows that it is a principal ideal domain and therefore a unique factorization domain.
- Theorem ZI2EUC. The domain is a Euclidean domain in which the absolute value of the norm is a suitable Euclidean function. Given any two nonzero numbers , it is always possible to find two other numbers such that , and .
- Proof. If is a divisor of , or the other way around, then or as needed, and . If this is not the case, it does not automatically mean and are coprime, but it does mean that but . Notate and . The we have:
- where . Now choose such that and then set and . Since the norm is multiplicative, it follows that and therefore
- as specified by the theorem. □
Representation of 2 in various bases
Obviously, in binary, 2 is represented as 10. For all ordinary higher integer bases, 2 is 2. In the balanced ternary numeral system, 2 is {1, −1}, meaning . In negabinary, 2 is 110, since . In quater-imaginary base, 2 is 2. In both the factorial numeral system and in -base, 2 is 100. And in the phi numeral system, 2 is 10.01, since .
See also
References
- ↑ This is Theorem 9.29 (4) in Niven-Zuckerman. See also Theorem P2 in quadratic integer rings#Primes in quadratic integer rings.
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 |