17 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical coincidence | 3/5 | https://en.wikipedia.org/wiki/Mathematical_coincidence | reference | science, encyclopedia | 2026-05-05T07:23:33.671306+00:00 | kb-cron |
π
9
/
e
8
≈
10
{\displaystyle \pi ^{9}/e^{8}\approx 10}
, to about 5 decimal places. That is,
ln
(
π
)
≈
ln
(
10
)
+
8
9
{\displaystyle \ln(\pi )\approx {\ln(10)+8 \over 9}}
, within 0.0002%.
2
π
+
e
≈
9
{\displaystyle 2\pi +e\approx 9}
, within 0.02%.
e
−
π
9
+
e
−
4
π
9
+
e
−
9
π
9
+
e
−
16
π
9
+
e
−
25
π
9
+
e
−
36
π
9
+
e
−
49
π
9
+
e
−
64
π
9
=
1.00000000000105
…
≈
1
{\textstyle e^{-{\frac {\pi }{9}}}+e^{-4{\frac {\pi }{9}}}+e^{-9{\frac {\pi }{9}}}+e^{-16{\frac {\pi }{9}}}+e^{-25{\frac {\pi }{9}}}+e^{-36{\frac {\pi }{9}}}+e^{-49{\frac {\pi }{9}}}+e^{-64{\frac {\pi }{9}}}=1.00000000000105\ldots \approx 1}
. In fact, this generalizes to the approximate identity
∑
k
=
1
n
−
1
e
−
k
2
π
n
≈
−
1
+
n
2
,
{\displaystyle \textstyle \sum _{k=1}^{n-1}{e^{-{\frac {k^{2}\pi }{n}}}}\approx {\frac {-1+{\sqrt {n}}}{2}},}
which can be explained by the Jacobian theta functional identity. Ramanujan's constant:
e
π
163
≈
262537412640768744
=
12
3
(
231
2
−
1
)
3
+
744
{\displaystyle e^{\pi {\sqrt {163}}}\approx 262537412640768744=12^{3}(231^{2}-1)^{3}+744}
, within
2.9
⋅
10
−
28
%
{\displaystyle 2.9\cdot 10^{-28}\%}
, discovered in 1859 by Charles Hermite. This very close approximation is not a typical sort of accidental mathematical coincidence, where no mathematical explanation is known or expected to exist (as is the case for most). It is a consequence of the fact that 163 is a Heegner number. There are several integers
k
=
2198
,
422151
,
614552
,
2508952
,
6635624
,
199148648
,
…
{\displaystyle k=2198,422151,614552,2508952,6635624,199148648,\dots }
((sequence A019297 in the OEIS)) such that
π
≈
ln
(
k
)
n
{\displaystyle \pi \approx {\frac {\ln(k)}{\sqrt {n}}}}
for some integer n, or equivalently
k
≈
e
π
n
{\displaystyle k\approx e^{\pi {\sqrt {n}}}}
for the same
n
=
6
,
17
,
18
,
22
,
25
,
37
,
…
{\displaystyle n=6,17,18,22,25,37,\dots }
These are not strictly coincidental because they are related to both Ramanujan's constant above and the Heegner numbers. For example,
k
=
199148648
=
14112
2
+
104
,
{\displaystyle k=199148648=14112^{2}+104,}
so these integers k are near-squares or near-cubes and note the consistent forms for n = 18, 22, 37,
π
≈
ln
(
784
2
−
104
)
18
{\displaystyle \pi \approx {\frac {\ln(784^{2}-104)}{\sqrt {18}}}}
π
≈
ln
(
1584
2
−
104
)
22
{\displaystyle \pi \approx {\frac {\ln(1584^{2}-104)}{\sqrt {22}}}}
π
≈
ln
(
14112
2
+
104
)
37
{\displaystyle \pi \approx {\frac {\ln(14112^{2}+104)}{\sqrt {37}}}}
with the last accurate to 13 decimal places.
(
e
e
)
e
≈
1000
φ
{\displaystyle (e^{e})^{e}\approx 1000\varphi }
10
(
e
π
−
ln
3
)
ln
2
=
318.000000033
…
{\displaystyle {\frac {10(e^{\pi }-\ln 3)}{\ln 2}}=318.000000033\ldots }
is almost an integer, to the 7th decimal place.
=== Other numerical curiosities === In a discussion of the birthday problem, the number
λ
=
1
365
(
23
2
)
=
253
365
{\displaystyle \lambda ={\frac {1}{365}}{23 \choose 2}={\frac {253}{365}}}
occurs, which is "amusingly" equal to
ln
(
2
)
{\displaystyle \ln(2)}
to 4 digits.
5
⋅
10
5
−
1
=
31
⋅
127
⋅
127
{\displaystyle 5\cdot 10^{5}-1=31\cdot 127\cdot 127}
, the product of three Mersenne primes.
6
!
6
{\displaystyle {\sqrt[{6}]{6!}}}
, the geometric mean of the first 6 natural numbers, is approximately 2.99; that is,
6
!
=
720
≈
729
=
3
6
{\displaystyle 6!=720\approx 729=3^{6}}
. The sixth harmonic number,
H
6
=
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
=
49
20
=
2.45
{\displaystyle H_{6}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}={\frac {49}{20}}=2.45}
which is approximately
6
{\displaystyle {\sqrt {6}}}
(2.449489...) to within 5.2 × 10−4.
109
5
≈
23
9
{\displaystyle {\sqrt[{5}]{109}}\approx {\frac {23}{9}}}
, within
2
×
10
−
7
{\displaystyle 2\times 10^{-7}}
.
=== Decimal coincidences ===
3
3
+
4
4
+
3
3
+
5
5
=
3435
{\displaystyle 3^{3}+4^{4}+3^{3}+5^{5}=3435}
, making 3435 the only non-trivial Münchhausen number in base 10 (excluding 0 and 1). If one adopts the convention that
0
0
=
0
{\displaystyle 0^{0}=0}
, however, then 438579088 is another Münchhausen number.
1
!
+
4
!
+
5
!
=
145
{\displaystyle \,1!+4!+5!=145}
and
4
!
+
0
!
+
5
!
+
8
!
+
5
!
=
40585
{\displaystyle \,4!+0!+5!+8!+5!=40585}
are the only non-trivial factorions in base 10 (excluding 1 and 2).
16
64
=
1
⧸
6
⧸
64
=
1
4
{\displaystyle {\frac {16}{64}}={\frac {1\!\!\!\not 6}{\not 64}}={\frac {1}{4}}}
,
26
65
=
2
⧸
6
⧸
65
=
2
5
{\displaystyle {\frac {26}{65}}={\frac {2\!\!\!\not 6}{\not 65}}={\frac {2}{5}}}
,
19
95
=
1
⧸
9
⧸
95
=
1
5
{\displaystyle {\frac {19}{95}}={\frac {1\!\!\!\not 9}{\not 95}}={\frac {1}{5}}}
, and
49
98
=
4
⧸
9
⧸
98
=
4
8
{\displaystyle {\frac {49}{98}}={\frac {4\!\!\!\not 9}{\not 98}}={\frac {4}{8}}}
. If the end result of these four anomalous cancellations are multiplied, their product reduces to exactly 1/100.
(
4
+
9
+
1
+
3
)
3
=
4913
{\displaystyle \,(4+9+1+3)^{3}=4913}
,
(
5
+
8
+
3
+
2
)
3
=
5832
{\displaystyle \,(5+8+3+2)^{3}=5832}
, and
(
1
+
9
+
6
+
8
+
3
)
3
=
19683
{\displaystyle \,(1+9+6+8+3)^{3}=19683}
. (In a similar vein,
(
3
+
4
)
3
=
343
{\displaystyle \,(3+4)^{3}=343}
.)
−
1
+
2
7
=
127
{\displaystyle \,-1+2^{7}=127}
, making 127 the smallest nice Friedman number. A similar example is
2
5
⋅
9
2
=
2592
{\displaystyle 2^{5}\cdot 9^{2}=2592}
.