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5
5 is a prime number, the only one to be both the largest member of a twin prime pair (with 3) and the smallest member of a twin prime pair (with 7). 5 is a factor of 10, our base of numeration, and was for a time a contender for base.
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 5
- 3 Partitions of 5
- 4 Roots and powers of 5
- 5 Logarithms and fifth powers
- 6 Values for number theoretic functions with 5 as an argument
- 7 Factorization of some small integers in a quadratic integer ring adjoining the square roots of −5, 5
- 8 Factorization of 5 in some quadratic integer rings
- 9 Representation of 5 in various bases
- 10 See also
- 11 References
Membership in core sequences
Odd numbers | 1, 3, 5, 7, 9, 11, 13, 15, 17, ... | A005408 |
Prime numbers | 2, 3, 5, 7, 11, 13, 17, 19, ... | A000040 |
Fibonacci numbers | 1, 1, 2, 3, 5, 8, 13, 21, ... | A000045 |
Partition numbers | 1, 1, 2, 3, 5, 7, 11, 15, 22, ... | A000041 |
In Pascal's triangle, 5 occurs twice, corresponding to and . (In Lozanić's triangle, 5 occurs four times).
Sequences pertaining to 5
Multiples of 5 | 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, ... | A008598 |
Pentagonal numbers | 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, ... | A000326 |
Generalized pentagonal numbers | 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, ... | A001318 |
Pentagonal pyramidal numbers | 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, ... | A002411 |
Centered pentagonal numbers | 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, ... | A005891 |
Fermat pseudoprimes to base 5 | 4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, ... | A005936 |
sequence starting at 3 | 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ... | A033478 |
sequence starting at 5 | 5, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, ... | A259207 |
Partitions of 5
There are seven partitions of 5, of which three consist of distinct numbers: {1, 4}, {2, 3} and {5}. There is only one nontrivial partition of 5 into primes, and that's {2, 3}.
Roots and powers of 5
In the table below, irrational numbers are given truncated to eight decimal places.
2.23606797 | A002163 | 5 2 | 25 | |
1.70997594 | A005481 | 5 3 | 125 | |
1.49534878 | A011003 | 5 4 | 625 | |
1.37972966 | A005534 | 5 5 | 3125 | |
1.30766048 | A011200 | 5 6 | 15625 | |
1.25849895 | A011201 | 5 7 | 78125 | |
1.22284454 | A011202 | 5 8 | 390625 | |
1.19581317 | A011203 | 5 9 | 1953125 | |
1.17461894 | A011204 | 5 10 | 9765625 | |
1.15755791 | A011205 | 5 11 | 48828125 | |
1.14352983 | A011206 | 5 12 | 244140625 | |
A000351 |
Logarithms and fifth powers
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 11, 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.
0.43067655 | A152675 | 2.32192809 | A020858 | 2 5 | 32 | |||
0.62133493 | 1.60943791 | A016628 | 148.41315910 | A092511 | ||||
0.68260619 | A152914 | 1.46497352 | A113209 | 3 5 | 243 | |||
0.71126066 | 1.40595430 | 306.01968478 | A092731 | |||||
0.86135311 | A153101 | 1.16096404 | A153201 | 4 5 | 1024 | |||
1.00000000 | 5 5 | 3125 | ||||||
1.11328275 | A153461 | 0.89824440 | A153202 | 6 5 | 7776 | |||
1.20906195 | A153616 | 0.82708747 | A153203 | 7 5 | 16807 | |||
1.29202967 | A153739 | 0.77397603 | A153204 | 8 5 | 32768 | |||
1.36521238 | A154008 | 0.73248676 | A153205 | 9 5 | 59049 | |||
1.43067655 | A154156 | 0.69897000 | A153268 | 10 5 | 100000 |
(See A000584 for the fifth powers of integers).
Values for number theoretic functions with 5 as an argument
–1 | ||
–2 | ||
3 | ||
6 | ||
2 | ||
4 | ||
1 | ||
1 | ||
4 | This is the Carmichael lambda function. | |
–1 | This is the Liouville lambda function. | |
1.036927755143369926331... (see A013663). | ||
5! | 120 | |
24 |
Factorization of some small integers in a quadratic integer ring adjoining the square roots of −5, 5
has quite a few things in common with . Both are actually subdomains because neither is integrally closed. Since (and likewise for –3), it is not enough to consider algebraic integers of the form (with ); we must also consider algebraic integers of the form (with ).
As it turns out, is an algebraic integer in . If this number looks familiar, that's probably because it's the golden ratio, . This means that , just as . And furthermore, just as is a unit in , so is a unit in , and both are unique factorization domains.
But then we come to a contrast: has just six units, but all the infinitely many powers of with integer exponents are units in (and that includes ). So, where a given number may seem to have two distinct factorizations in , these can be shown to be the same by multiplying or dividing by the appropriate power of .
also contrasts with , which is not a UFD but has only two units: 1 and –1. This allows us to be very certain when asserting that a given number has two distinct factorizations in .
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 rational and an irrational part.
1 | Unit | |
2 | Irreducible | Prime |
3 | ||
4 | 2 2 | |
5 | ||
6 | 2 × 3 OR | 2 × 3 |
7 | Irreducible | Prime |
8 | 2 3 | |
9 | 3 2 OR ^{[1]} | 3 2 |
10 | ||
11 | Prime | |
12 | 2 2 × 3 | |
13 | Prime | |
14 | 2 × 7 OR | 2 × 7 |
15 | ||
16 | 2 4 | |
17 | Prime | |
18 | 2 × 3 2 | |
19 | Prime | |
20 |
Of course nothing actually forbids us from regarding as a domain in its own right, even though it's not integrally closed. As it turns out, it does not have unique factorization, since, for example, . This fact may seem incompatible with being a UFD, but there is no conflict: the latter factorization is equivalent to which can be rewritten as which leads to before finally boiling down to 2 2 after all the units are stripped away.
It is worth noting that has class number 2. What this means is that a number may have more than one distinct factorization into irreducibles, but in every case each of those factorizations will have the same amount of irreducibles. For example, , wherein we see that both factorizations consist of two irreducibles each.
Now, the number 30 may seem like a counterexample, given that . But that's kind of like saying that 5 × 6 is a distinct factorization of 30, because neither 5 nor 6 is irreducible in . So, in order to prove that is not a distinct factorization of 30 consisting of only two irreducibles rather than four, we need to find two numbers with a norm of 5 and another two with a norm of 6. Naturally this leads us to find and , and we verify that .
Therefore, the two distinct factorizations of 30 are and . In one of the factorizations we have felt it necessary to include the unit , though we could certainly avoid this, if we so wished, by "shopping" the associates. And if in the other factorization we were so inclined, we could prefix "". But aside from this fiddling with the units, we essentially have two distinct factorizations each nevertheless consisting of four irreducibles.
It should be quite clear at this point that is not a Euclidean domain, since it's not a unique factorization domain. Of course that doesn't mean that the Euclidean algorithm will always fail to find the greatest common divisor of a given pair of numbers. Presumably if both and are purely real integers the Euclidean algorithm will still find .
What if is a purely real integer but is complex? Let's try . Since and , perhaps we can find numbers and in this domain such that , with and .
However, the four multiples of 2 nearest to all give such that , as the following table shows:
2 | 6 | |
6 | ||
6 | ||
0 | 6 |
The intuition here is that as we pick multiples of 2 that are further away from , the norm of the remainder will be increasingly larger.
But by using the basic property of parity, we need not rely on an intuition. If , with , it is obvious that both and are even. Then we have .
Since and either, it follows that 6 is the minimum norm of . Hence the Euclidean algorithm can't proceed because it can't even start.
Factorization of 5 in some quadratic integer rings
In , 5 is a prime number. But it has different factorizations in some quadratic integer rings.
Prime | Prime | ||
Irreducible | |||
Prime | Prime | ||
Irreducible | Irreducible | ||
Irreducible | Prime | ||
Irreducible | |||
Prime | |||
. |
We can say with certainty that 5 is prime in , and . For any other imaginary quadratic rings, we can only say it is irreducible, at least not without some examination of ideals in a specific ring.
Representation of 5 in various bases
Base 2 3 4 5 6 through 36 Representation 101 12 11 10 5
In the balanced ternary numeral system, 5 is {1, –1, –1}, meaning . In negabinary, 5 has the same representation as in binary. In quater-imaginary base, 10 is 10301, since . In the factorial numeral system, 5 is 21, since .
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 |
References
- ↑ John J. Watkins, Topics in Commutative Ring Theory. Princeton and Oxford: Princeton University Press (2007): 91