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4
4 is the square of 2.
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 4
- 3 Partitions of 4
- 4 Roots and powers of 4
- 5 Logarithms and fourth powers
- 6 Values for number theoretic functions with 4 as an argument
- 7 Factorization of 4 in some quadratic integer rings
- 8 Representation of 4 in various bases
- 9 See also
- 10 References
Membership in core sequences
Even numbers | 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ... | A005843 |
Composite numbers | 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, ... | A002808 |
Powers of 2 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... | A000079 |
Perfect squares | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ... | A000290 |
Lucas numbers | 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... | A000032 |
In Pascal's triangle, 4 occurs twice, surrounding the first instance of 6. (In Lozanić's triangle, 4 occurs five times).
Sequences pertaining to 4
Multiples of 4 | 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ... | A008586 |
Centered square numbers | 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... | A001844 |
Partitions of 4
There are five partitions of 4, only one of which consists of distinct numbers: {1, 3}. The only possible prime partition is {2, 2}.
Roots and powers of 4
Since 4 = 2 2, it follows from the basic properties of exponentiation that the powers of 4 are the even-indexed powers of 2. Whereas the powers of 2 are congruent to 2 or 1 modulo 3, all powers of 4 are congruent to 1 modulo 3. One consequence of this concerns the Collatz function: the powers of 4 can be accessed either from a halving step or from a "tripling" step, while all other powers of 2 can only be accessed from a halving step.
Another fact readily verified by the basic properties of exponentiation is that .
In the table below, irrational numbers are given truncated to eight decimal places.
2.00000000 | A000038 | 4 2 | 16 | |
1.58740105 | A005480 | 4 3 | 64 | |
1.41421356 | A002193 | 4 4 | 256 | |
1.31950791 | A005533 | 4 5 | 1024 | |
1.25992104 | A002580 | 4 6 | 4096 | |
1.21901365 | A011186 | 4 7 | 16384 | |
1.18920711 | A010767 | 4 8 | 65536 | |
1.16652903 | A011188 | 4 9 | 262144 | |
1.14869835 | A005531 | 4 10 | 1048576 | |
1.13431252 | A011190 | 4 11 | 4194304 | |
1.12246204 | A010768 | 4 12 | 16777216 | |
1.11253147 | A011192 | 4 13 | 67108864 | |
1.10408951 | A010769 | 4 14 | 268435456 | |
1.09682497 | A011194 | 4 15 | 1073741824 | |
1.09050773 | A010770 | 4 16 | 4294967296 | |
A000302 |
Logarithms and fourth powers
In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript.
From the basic properties of exponentiation, it follows that all fourth powers are squares, since . Hence fourth powers are sometimes called "biquadrates." And from Fermat's little theorem it follows that if is coprime to 5, then .
As above, irrational numbers in the following table are truncated to eight decimal places.
0.50000000 | A020761 | 2.00000000 | 2 4 | 16 | ||||
0.72134752 | A133362 | 1.38629436 | A016627 | 54.59815003 | A092426 | |||
0.79248125 | A094148 | 1.26185950 | A100831 | 3 4 | 81 | |||
0.82574806 | 1.21102312 | 97.40909103 | A092425 | |||||
1.00000000 | 4 4 | 256 | ||||||
1.16096404 | A153201 | 0.86135311 | A153101 | 5 4 | 625 | |||
1.29248125 | A153460 | 0.77370561 | A153102 | 6 4 | 1296 | |||
1.40367746 | A153615 | 0.71241437 | A153103 | 7 4 | 2401 | |||
1.50000000 | 0.66666666 | 8 4 | 4096 | |||||
1.58496250 | A020857 | 0.63092975 | A102525 | 9 4 | 6561 | |||
1.66096404 | A154155 | 0.60205999 | A114493 | 10 4 | 10000 |
(See A000583 for the fourth powers of integers).
Values for number theoretic functions with 4 as an argument
0 | ||
–1 | ||
2 | ||
7 | ||
3 | ||
2 | ||
2 | ||
1 | ||
2 | This is the Carmichael lambda function. | |
1 | This is the Liouville lambda function. | |
1.082323233711138191516... (see A013662). | ||
4! | 24 | |
6 |
Factorization of 4 in some quadratic integer rings
As was mentioned above, 4 is the square of 2 in . But it has different factorizations in some quadratic integer rings. We could just take the factorizations of 2 and stick in a bunch of 2s as exponents, change some exponent 2s to exponent 4s. That works, at least for those rings that are unique factorization domains, but it seems to ignore rings that are not UFDs.
And yet, it is possible for an integer that is composite in to still have only one factorization in a non-UFD. For example, one might think that constitutes a factorization of 4 in apart from 2 2, but it isn't, since is a unit and . A very similar thing happens in : suffice it to say that is a unit... In general, if both parts are even, the factorization is probably not distinct.
The table below would have to go at least up to 65 to show a real quadratic ring in which 4 has more than one distinct factorization: in , 2 is irreducible, yet .
Despite the fact that very few imaginary quadratic rings are UFDs, 4 has only one distinct factorization in all imaginary rings except one, namely , as the table below shows. In all other imaginary rings besides the ones listed below, 2 is irreducible and consequently 4 can only be factored as 2 2.
2 2 | |||
2 2 | |||
2 2 | 2 2 | ||
2 2 | |||
2 2 OR | 2 2 | ||
2 2 | |||
Representation of 4 in various bases
Base 2 3 4 5 through 36 Representation 100 11 10 4
The negabinary representation of 4 is the same as in binary, and its balanced ternary numeral system representation is the same as its ternary. In -base, 4 is 10000. The quater-imaginary base representation of 4 is somewhat more interesting, being 10300, since .
The table above shows 4 is a palindromic number in ternary, and trivially palindromic in all higher integer bases. And it turns out that 4 is a Harshad number in all positional positive integer bases.
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 |