Number that is its own Gödel number
In number theory and mathematical logic , a Meertens number in a given number base
b
{\displaystyle b}
is a natural number that is its own Gödel number . It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI , Amsterdam .[ 1]
Definition
Let
n
{\displaystyle n}
be a natural number. We define the Meertens function for base
b
>
1
{\displaystyle b>1}
F
b
:
N
→ → -->
N
{\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }
to be the following:
F
b
(
n
)
=
∏ ∏ -->
i
=
0
k
− − -->
1
p
k
− − -->
i
− − -->
1
d
i
.
{\displaystyle F_{b}(n)=\prod _{i=0}^{k-1}p_{k-i-1}^{d_{i}}.}
where
k
=
⌊ ⌊ -->
log
b
-->
n
⌋ ⌋ -->
+
1
{\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}
is the number of digits in the number in base
b
{\displaystyle b}
,
p
i
{\displaystyle p_{i}}
is the
i
{\displaystyle i}
-prime number , and
d
i
=
n
mod
b
i
+
1
− − -->
n
mod
b
i
b
i
{\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}
is the value of each digit of the number. A natural number
n
{\displaystyle n}
is a Meertens number if it is a fixed point for
F
b
{\displaystyle F_{b}}
, which occurs if
F
b
(
n
)
=
n
{\displaystyle F_{b}(n)=n}
. This corresponds to a Gödel encoding .
For example, the number 3020 in base
b
=
4
{\displaystyle b=4}
is a Meertens number, because
3020
=
2
3
3
0
5
2
7
0
{\displaystyle 3020=2^{3}3^{0}5^{2}7^{0}}
.
A natural number
n
{\displaystyle n}
is a sociable Meertens number if it is a periodic point for
F
b
{\displaystyle F_{b}}
, where
F
b
k
(
n
)
=
n
{\displaystyle F_{b}^{k}(n)=n}
for a positive integer
k
{\displaystyle k}
, and forms a cycle of period
k
{\displaystyle k}
. A Meertens number is a sociable Meertens number with
k
=
1
{\displaystyle k=1}
, and a amicable Meertens number is a sociable Meertens number with
k
=
2
{\displaystyle k=2}
.
The number of iterations
i
{\displaystyle i}
needed for
F
b
i
(
n
)
{\displaystyle F_{b}^{i}(n)}
to reach a fixed point is the Meertens function's persistence of
n
{\displaystyle n}
, and undefined if it never reaches a fixed point.
Meertens numbers and cycles of Fb for specific b
All numbers are in base
b
{\displaystyle b}
.
b
{\displaystyle b}
Meertens numbers
Cycles
Comments
2
10, 110, 1010
n
<
2
96
{\displaystyle n<2^{96}}
[ 2]
3
101
11 → 20 → 11
n
<
3
60
{\displaystyle n<3^{60}}
[ 2]
4
3020
2 → 10 → 2
n
<
4
48
{\displaystyle n<4^{48}}
[ 2]
5
11, 3032000, 21302000
n
<
5
41
{\displaystyle n<5^{41}}
[ 2]
6
130
12 → 30 → 12
n
<
6
37
{\displaystyle n<6^{37}}
[ 2]
7
202
n
<
7
34
{\displaystyle n<7^{34}}
[ 2]
8
330
n
<
8
32
{\displaystyle n<8^{32}}
[ 2]
9
7810000
n
<
9
30
{\displaystyle n<9^{30}}
[ 2]
10
81312000
n
<
10
29
{\displaystyle n<10^{29}}
[ 2]
11
∅ ∅ -->
{\displaystyle \varnothing }
n
<
11
44
{\displaystyle n<11^{44}}
[ 2]
12
∅ ∅ -->
{\displaystyle \varnothing }
n
<
12
40
{\displaystyle n<12^{40}}
[ 2]
13
∅ ∅ -->
{\displaystyle \varnothing }
n
<
13
39
{\displaystyle n<13^{39}}
[ 2]
14
13310
n
<
14
25
{\displaystyle n<14^{25}}
[ 2]
15
∅ ∅ -->
{\displaystyle \varnothing }
n
<
15
37
{\displaystyle n<15^{37}}
[ 2]
16
12
2 → 4 → 10 → 2
n
<
16
24
{\displaystyle n<16^{24}}
[ 2]
See also
References
External links
Possessing a specific set of other numbers
Expressible via specific sums