--- title: "Mathematical coincidence" chunk: 3/5 source: "https://en.wikipedia.org/wiki/Mathematical_coincidence" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T07:23:33.671306+00:00" instance: "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} .