# Archive for May 26th, 2015

## The Fundamental theorem of arithmetic – SQL version

Posted by Matthias Rogel on 26. May 2015

Every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes, see for example Wolfram MathWorld or Wikipedia.

Here is the SQL-Version, we compute this for all integers up to 100

```with bound as
(
select 100 as bound from dual
),
n_until_bound as (
select level+1 n
from dual
connect by level <= (select bound.bound from bound)
),
primes_under_bound as
(
select n_until_bound.n as prime
from n_until_bound
minus
select n1.n * n2.n
from n_until_bound n1, n_until_bound n2
where n1.n <= n2.n
and n1.n <= (select sqrt(bound.bound) from bound)
),
primepowers_until_bound as
(
select p.prime, l.exponent
from primes_under_bound p, lateral(select level as exponent from dual connect by level <= log(p.prime, (select bound.bound from bound))) l
),
factors as
(
select n.n, pb.prime, pb.exponent
from n_until_bound n, primepowers_until_bound pb
where mod(n.n, power(pb.prime, pb.exponent)) = 0
),
largestfactors as
(
select
f.n, f.prime, min(f.exponent) keep(dense_rank first order by f.exponent desc) as exponent
from factors f
group by f.n, f.prime
)
select /*+pallel */ lf.n || ' = ' || listagg(lf.prime || case when lf.exponent > 1 then ' ^ ' || lf.exponent end, ' * ') within group(order by lf.prime asc) as factorization
from largestfactors lf
group by lf.n
order by lf.n
FACTORIZATION
----------------------------------------------------------------------------------------------------
2 = 2
3 = 3
4 = 2 ^ 2
5 = 5
6 = 2 * 3
7 = 7
8 = 2 ^ 3
9 = 3 ^ 2
10 = 2 * 5
11 = 11
12 = 2 ^ 2 * 3
13 = 13
14 = 2 * 7
15 = 3 * 5
16 = 2 ^ 4
17 = 17
18 = 2 * 3 ^ 2
19 = 19
20 = 2 ^ 2 * 5
21 = 3 * 7
22 = 2 * 11
23 = 23
24 = 2 ^ 3 * 3
25 = 5 ^ 2
26 = 2 * 13
27 = 3 ^ 3
28 = 2 ^ 2 * 7
29 = 29
30 = 2 * 3 * 5
31 = 31
32 = 2 ^ 5
33 = 3 * 11
34 = 2 * 17
35 = 5 * 7
36 = 2 ^ 2 * 3 ^ 2
37 = 37
38 = 2 * 19
39 = 3 * 13
40 = 2 ^ 3 * 5
41 = 41
42 = 2 * 3 * 7
43 = 43
44 = 2 ^ 2 * 11
45 = 3 ^ 2 * 5
46 = 2 * 23
47 = 47
48 = 2 ^ 4 * 3
49 = 7 ^ 2
50 = 2 * 5 ^ 2
51 = 3 * 17
52 = 2 ^ 2 * 13
53 = 53
54 = 2 * 3 ^ 3
55 = 5 * 11
56 = 2 ^ 3 * 7
57 = 3 * 19
58 = 2 * 29
59 = 59
60 = 2 ^ 2 * 3 * 5
61 = 61
62 = 2 * 31
63 = 3 ^ 2 * 7
64 = 2 ^ 6
65 = 5 * 13
66 = 2 * 3 * 11
67 = 67
68 = 2 ^ 2 * 17
69 = 3 * 23
70 = 2 * 5 * 7
71 = 71
72 = 2 ^ 3 * 3 ^ 2
73 = 73
74 = 2 * 37
75 = 3 * 5 ^ 2
76 = 2 ^ 2 * 19
77 = 7 * 11
78 = 2 * 3 * 13
79 = 79
80 = 2 ^ 4 * 5
81 = 3 ^ 4
82 = 2 * 41
83 = 83
84 = 2 ^ 2 * 3 * 7
85 = 5 * 17
86 = 2 * 43
87 = 3 * 29
88 = 2 ^ 3 * 11
89 = 89
90 = 2 * 3 ^ 2 * 5
91 = 7 * 13
92 = 2 ^ 2 * 23
93 = 3 * 31
94 = 2 * 47
95 = 5 * 19
96 = 2 ^ 5 * 3
97 = 97
98 = 2 * 7 ^ 2
99 = 3 ^ 2 * 11
100 = 2 ^ 2 * 5 ^ 2

```