11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical coincidence | 4/5 | https://en.wikipedia.org/wiki/Mathematical_coincidence | reference | science, encyclopedia | 2026-05-05T07:23:33.671306+00:00 | kb-cron |
1
3
+
5
3
+
3
3
=
153
{\displaystyle \,1^{3}+5^{3}+3^{3}=153}
,
3
3
+
7
3
+
0
3
=
370
{\displaystyle \,3^{3}+7^{3}+0^{3}=370}
,
3
3
+
7
3
+
1
3
=
371
{\displaystyle \,3^{3}+7^{3}+1^{3}=371}
, and
4
3
+
0
3
+
7
3
=
407
{\displaystyle \,4^{3}+0^{3}+7^{3}=407}
are all narcissistic numbers.
588
2
+
2353
2
=
5882353
{\displaystyle \,588^{2}+2353^{2}=5882353}
, a prime number. The fraction 1/17 also produces 0.05882353 when rounded to 8 digits.
2
1
+
6
2
+
4
3
+
6
4
+
7
5
+
9
6
+
8
7
=
2646798
{\displaystyle \,2^{1}+6^{2}+4^{3}+6^{4}+7^{5}+9^{6}+8^{7}=2646798}
. The largest number with this pattern is
12157692622039623539
=
1
1
+
2
2
+
1
3
+
…
+
9
20
{\displaystyle \,12157692622039623539=1^{1}+2^{2}+1^{3}+\ldots +9^{20}}
.
13532385396179
=
13
×
53
2
×
3853
×
96179
{\displaystyle 13532385396179=13\times 53^{2}\times 3853\times 96179}
. This number, found in 2017, answers a question by John Conway whether the digits of a composite number could be the same as its prime factorization. A similar example (in fact the smallest) in binary is
255987
=
3
3
×
19
×
499
{\displaystyle 255987=3^{3}\times 19\times 499}
, whose prime factorization in binary reads
111110011111110011
=
11
11
×
10011
×
111110011
{\displaystyle 111110011111110011=11^{11}\times 10011\times 111110011}
. Examples in other bases include
3518
=
3
5
×
18
{\displaystyle 3518=3^{5}\times 18}
in base 11 (in decimal:
4617
=
3
5
×
19
{\displaystyle 4617=3^{5}\times 19}
), and
15287
=
15
2
×
87
{\displaystyle 15287=15^{2}\times 87}
in base 12 (in decimal:
29767
=
17
2
×
103
{\displaystyle 29767=17^{2}\times 103}
). Also, there exist pairs of numbers such that each is the concatenation of the primes and exponents in the prime factorization of the other in binary:
1111101111
=
10011
×
110101
{\displaystyle 1111101111=10011\times 110101}
(
1007
=
19
×
53
{\displaystyle 1007=19\times 53}
),
10011110101
=
11
11
×
101111
{\displaystyle 10011110101=11^{11}\times 101111}
(
1269
=
3
3
×
47
{\displaystyle 1269=3^{3}\times 47}
);
10111011111
=
11
10
×
10100111
{\displaystyle 10111011111=11^{10}\times 10100111}
(
1503
=
3
2
×
167
{\displaystyle 1503=3^{2}\times 167}
),
111010100111
=
1011
10
×
11111
{\displaystyle 111010100111=1011^{10}\times 11111}
(
3751
=
11
2
×
31
{\displaystyle 3751=11^{2}\times 31}
).
=== Numerical coincidences in numbers from the physical world ===
==== Speed of light ==== The speed of light is (by definition) exactly 299792458 m/s, extremely close to 3.0×108 m/s (300000000 m/s). This is a pure coincidence, as the metre was originally defined as 1 / 10000000 of the distance between the Earth's pole and equator along the surface at sea level, and the Earth's circumference just happens to be about 2/15 of a light-second. It is also roughly equal to one foot per nanosecond (the actual number is 0.9836 ft/ns).
==== Angular diameters of the Sun and the Moon ==== As seen from Earth, the angular diameter of the Sun varies between 31′27″ and 32′32″, while that of the Moon is between 29′20″ and 34′6″. The fact that the intervals overlap (the former interval is contained in the latter) is a coincidence, and has implications for the types of solar eclipses that can be observed from Earth.
==== Gravitational acceleration ====
While not constant but varying depending on latitude and altitude, the numerical value of the acceleration caused by Earth's gravity on the surface lies between 9.74 and 9.87 m/s2, which is quite close to 10. This means that as a result of Newton's second law, the weight of a kilogram of mass on Earth's surface corresponds roughly to 10 newtons of force exerted on an object. This is related to the aforementioned coincidence that the square of pi is close to 10. One of the early definitions of the metre was the length of a pendulum whose half swing had a period equal to one second. Since the period of the full swing of a pendulum is approximated by the equation below, algebra shows that if this definition was maintained, gravitational acceleration measured in metres per second per second would be exactly equal to π2.
T
≈
2
π
L
g
{\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}}
The upper limit of gravity on Earth's surface (9.87 m/s2) is equal to π2 m/s2 to four significant figures. It is approximately 0.6% greater than standard gravity (9.80665 m/s2).
==== Rydberg constant ==== The Rydberg constant, when multiplied by the speed of light and expressed as a frequency, is close to
π
2
3
×
10
15
Hz
{\displaystyle {\frac {\pi ^{2}}{3}}\times 10^{15}\ {\text{Hz}}}
:
3.2898
_
41960364
(
17
)
×
10
15
Hz
=
R
∞
c
{\displaystyle {\underline {3.2898}}41960364(17)\times 10^{15}\ {\text{Hz}}=R_{\infty }c}
3.2898
_
68133696
…
=
π
2
3
{\displaystyle {\underline {3.2898}}68133696\ldots ={\frac {\pi ^{2}}{3}}}
This is also approximately the number of feet in one meter:
3.28084
{\displaystyle 3.28084}
ft
≈
1
{\displaystyle \approx 1}
m
==== US customary to metric conversions ==== One mile is the same length as 1.609344 kilometres. As discovered by Randall Munroe, a cubic mile is close to
4
3
π
{\displaystyle {\frac {4}{3}}\pi }
cubic kilometres (within 0.5%). This means that a sphere with radius n kilometres has almost exactly the same volume as a cube with side length n miles. Exact equality would make the conversion factor equal to
(
4
3
π
)
1
3
≈
1.612
.
{\displaystyle ({\tfrac {4}{3}}\pi )^{\frac {1}{3}}\approx 1.612\,.}