The number 163
The numbers π and e are famous constants, approximately equal to 3.141592653589793... and 2.718281828459045..., respectively. Remarkably, the number e to the power π√163 is within 0.000000000001 of an integer. This may seem like a coincidence, but it is not.
The key to the mystery is that the number 163 has some unusual properties in the context of unique factorization. An important and very useful property of the integers, called the Fundamental Theorem of Arithmetic, is that every integer greater than 1 can be factorized as a product of primes in an essentially unique way. For example, the integer 21 can be written as 3×7, or as 7×3, or as (−3)×(−7), or as (−7)×(−3). We consider these factorizations to be essentially the same, because they only differ in (a) the order of the factors and (b) multiplication by units. An integer z is a unit if 1/z is also an integer, which in this case means that z must be 1 or −1.
Another number system (or ring) that has the unique factorization property is the Gaussian integers. These are the complex numbers of the form a + bi, where a and b are integers, and i is a square root of −1. The Gausian integers can be thought of as square grid points in the plane, as in the picture above. The units in the Gaussian integers are the numbers {1, i, −1, −i}; this means that these are the only four Gaussian integers z with the property that 1/z is also a Gaussian integer. In the Gaussian integers, the number 5 is not prime: it can be factorized as 5=(2+i)×(2−i), or as 5=(1−2i)×(1+2i). These do not count as different factorizations because 1−2i is a unit multiple of 2+i (since (1−2i)=(−i)×(2+i)) and similarly 1+2i is a unit multiple of 2−i. The element 1−2i is a prime in the Gaussian integers, because there are no ways to write 1−2i as a product of two other Gaussian integers without one of the factors being a unit.
An example of a ring that does not have the unique factorization property is the ring Z[√(−5)]. This consists of the complex numbers of the form a + b√(−5), where √(−5)=√5i is a square root of −5. In this case, it turns out that there are two essentially different ways to factorize 6, namely 6=2×3, and 6=(1+√(−5))×(1−√(−5)).
The Eisenstein integers consist of the complex numbers of the form a + bω, where ω=(1+√(−3))/2. When plotted on the plane, these numbers are the grid points of a triangular lattice, as shown above. The Eisenstein integers have the unique factorization property, and the dark points in the picture denote the prime elements, including the numbers 2, 5, and 11 on the horizontal axis.
The ring of integers of Q[√d] consists of the complex numbers of the form a + bω, where ω is equal to √d unless d−1 is a multiple of 4, in which case ω is given by (1+√d)/2. In particular, the Gaussian integers are the ring of integers of Q[√(−1)]; the ring Z[√(−5)] is the ring of integers of Q[√(−5)]; and the Eisenstein integers are the ring of integers of Q[√(−3)]. This ring of integers has the unique factorization property if d is equal to −1 or −3, but not if d is equal to −5. The Baker–Heegner–Stark theorem gives the complete list of negative integers d for which the ring of integers of Q[√d] has the unique factorization property. There are only nine numbers on the list, which are the negatives of the numbers 1, 2, 3, 7, 11, 19, 43, 67, and 163. These numbers are called the Heegner numbers, and many of the remarkable properties of the number 163 derive from the fact that it is the largest Heegner number.
The mysterious connection between the numbers 163, e, and π relates to the j-function. A partial formula for this is given above; here q, is shorthand for e to the power 2πiτ. It is a theorem that if d is a Heegner number, then j((1+√(−d))/2) is an integer. In the special case where d=163, the number j((1+√(−163))/2) equals 640320 raised to the power 3. A calculation shows that in this case, −1/q is equal to e^(π√163), which is a big number. It follows that 1/q + 744 is an excellent approximation to j(τ), which explains why e to the power π√163 is so close to the integer (640320)^3 + 744, or 262537412640768744.
The connection between 163 and 640320 plays an important part in the above formula for π, which was found by the brothers David and Gregory Chudnovsky in 1987. In March 2022, this formula was used to calculate π to 100 trillion digits. The reason the formula is so effective is that it converges extremely quickly. Even though the formula is an infinite sum, using only the first term gives an approximate value of π of 3.141592653589734, which is already accurate to 13 decimal places.
The mysterious number 744 that shows up in the j-function also appears in the TV show The 100, where it is the number of symbols on the Anomaly Stone used for interstellar travel (shown above). The number 744 is also equal to 3×248, where 248 is the dimension of the simple Lie algebra E8, but it is not clear that there is any cosmic significance to this.
Picture credits and relevant links
Wikipedia has articles on the Gaussian integers, the Eisenstein integers, the j-function, the Baker–Heegner–Stark theorem, and the Heegner numbers.
The picture of the Gaussian integers comes from the paper Gaussian Integers and Arctangent Identities for π by Jack S. Calcut.
The picture of the Eisenstein primes is by Fropuff and appears on Wikipedia.
The picture of the Anomaly Stone comes from a fan site for The 100.
The other graphics are my own work.