This blog post is based on a Twitter thread on the same topic.
There is something about prime numbers… An air of mystery surrounds them, that makes them one of the most alluring (and most studied) objects in all of mathematics. Despite hundreds of years of prime number research, there is still so much we do not know about them. Of course, we know that there are infinitely many prime numbers, with a first proof due to Euclid and many, many other equally fascinating proofs that continue to be found. Nonetheless, many open problems about their distribution among the natural numbers remain wide open. The Riemann Hypothesis, for instance, is intimately intertwined with the distribution of prime numbers.
In addition to the mysterious nature of the prime numbers as a whole, certain individual primes have a special place in my heart, for various reasons. In this blog post, I will list a few of my favorite primes, together with the fascinating properties that make them special… to me! The reader and other mathematicians would certainly compose different lists of favorite primes.
Without further ado, the list begins with the very first of all prime numbers…
The number 2 is the first prime, the smallest prime, and the pain of number theorists’ existence. It is such an odd prime that there is no other quite like it (unless you look for prime ideals above 2 in other number fields other than Q!). All sorts of curious facts come back to the fact that 2 is the unique even prime. For example:
- If q=mn-1 is a prime number, for some natural number m, then either q=2 or m=2. Primes of the form q=2n-1 are called Mersenne primes (which will make another guest appearance below).
- See also the special role of 2 in the construction of Fermat numbers and Fermat primes below.
- For any n>1, a polygon with 2n sides can be constructed with a ruler and compass, but if you replace 2 by any other prime p, this is no longer true (we will come back to this point later on).
- The Law of Quadratic Reciprocity gives a beautiful relationship between pairs of primes, but the prime 2 is a complete outlier in this regard, and it does not behave at all like the rest of the primes.
- The group (Z/pnZ)x is cyclic for all primes p>2 and all n>0, but it is not cyclic for p=2 and n>2.
p=37 might be my all-time favorite prime, for silly reasons such as 37*3 = 111, 37*6 = 222,… , and also for deeper reasons such as the fact that 37 is the first irregular prime. The regular primes are those exponents for which Fermat’s last theorem has a “simple proof” (first discovered by Lamé, who proposed an erroneous proof of Fermat’s last theorem, which was later fixed by Kummer for regular primes). The irregular primes, 37, 59, 67, 101, 103, 131, 149,… are those for which Kummer’s proof doesn’t work. In particular, this means that the class group of the ring of integers of the 37th cyclotomic field is of order divisible by 37… and in this case it is exactly of order 37.
Another couple of reasons why I am fascinated by the number 37 come from the theory of elliptic curves. A map between two elliptic curves is called an isogeny, and it turns out that cyclic, rational isogenies are somewhat rare. The size of the kernel of the map is called the degree of the isogeny, and Barry Mazur showed that there are only finitely many primes that are degrees of isogenies of elliptic curves. As it turns out, p=37 is one of the degrees that can occur… but it only occurs for two (isomorphism classes of) elliptic curves (1225.b1 and 1225.b2), and these elliptic curves are rather special. The second reason will be explained below.
The prime number 163 is really nice for several reasons. For instance, epi*Sqrt(163) is really close to being an integer (it is 262537412640768743.99999999999925… so an integer to 12 decimal places) which has a very interesting explanation coming from elliptic curves with complex multiplication. Not completely unrelated to this the previous fact, Q(Sqrt(-163)) is the “last” of the imaginary quadratic fields of class number 1 (there are only nine such fields, and this is the one with largest discriminant in absolute value). And also in the same family of amazing facts: the values of the polynomial x2-x+41 for x=0 up to x=40 are prime numbers! Finally, 163 is the largest possible degree of a cyclic, rational isogeny for an elliptic curve defined over Q.
p=1093 and 3511
Fermat’s little theorem says that if p is an odd prime, then p is a divisor of the number 2(p-1) – 1. A Wieferich prime is a prime p such that p2 is a divisor of 2(p-1) – 1. We only know two Wieferich primes: 1093 and 3511. The crazy thing is that we conjecture that there are infinitely many Wieferich primes… but we only know two of them! More concretely, we expect log(log(x)) Wieferich primes below x, and since log(log(x)) grows so slowly, we are not surprised we haven’t found any others yet. I became interested in Wieferich primes (in fact, Wieferich places) when they unexpectedly showed up in some work of mine.
p=4001 and 4003
The twin prime conjecture claims that there are infinitely many natural numbers n such that n and n+2 are both primes. Sometimes, it is useful to have a “large” pair of twin primes to compute with, and 4001 and 4003 are easy to remember, large enough for most purposes, and not too large at the same time. That’s it. They are stuck in my head, and I use them very often!
p=11, 37, 389, 5077, (117223), and 19047851
The set E(Q) of all rational points on an elliptic curve E defined over Q is a finitely generated abelian group (thanks to the Mordell-Weil theorem), so E(Q) has a finite torsion subgroup T(E/Q), and also R(E/Q) rational points of infinite order such that E(Q) is isomorphic to T(E/Q) + ZR(E/Q). No one knows how large the rank R(E/Q) of an elliptic curve over Q can be, or what values R(E/Q) can take for that matter. The largest known rank is 28 (an example due to Noam Elkies). So it is interesting to find the “simplest” elliptic curves with any given rank. We organize elliptic curves by their conductor, so it is interesting to find examples of elliptic curves with rank R(E/Q)=0, 1, 2, 3, 4,… with the smallest possible conductor. Here is the beginning of such a list, with curves given by their LMFDB.org label:
- R(E/Q) = 0, conductor 11, curve 11.a1.
- R(E/Q) = 1, conductor 37, curve 37.a1.
- R(E/Q) = 2, conductor 389, curve 389.a1.
- R(E/Q) = 3, conductor 5077, curve 5077.a1.
- R(E/Q) = 4, conductor 234446 = 2*117223, curve 234446a1.
- R(E/Q) = 5, conductor 19047851, curve 19047851.a1.
The curves of rank 3 and conductor 5077 have a special place in the history of number theory, and 5077a1 is called the “Gauss curve” (see the paragraph at the bottom of this LMFDB page). As far as I know, there is an elliptic curve of rank 6 and conductor 5187563742=2*3*2777*311341 but it is not proven to be the smallest such conductor!
Even though we have a proof that there are infinitely many prime numbers, finding very large prime numbers is a very difficult task. Thus, it would be of great interest if there was a simple formula or function that produced prime numbers. One famous such “formula” was proposed by Fermat, who famously claimed that the numbers of the form Fn = 22^n+1, known as Fermat numbers, are always prime. The first few Fermat numbers F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 are, indeed, prime numbers. However, Fermat’s claim has been proven to be fantastically wrong, since every single other Fermat number that we have been able to factor has turned out to be a composite number. For instance, Euler proved in 1732 that F5 = 4294967297 = 641*6700417.
Fermat primes, if you can find them, are really cool, because of the Gauss-Wantzel theorem which says that a regular polygon with n sides can be constructed with a compass and ruler (straightedge, no markings) if and only if n is the product of a power of 2 and any number of distinct Fermat primes. So, in particular, there is a construction of a polygon with 65537 using just a compass and a ruler!
It should be obvious why I love this one! One can ask if there are palindromic numbers, with digits in order, that are prime. The sequence that I have in mind is 1, 121, 12321, 1234321, etc., and none of these numbers are prime, until you reach
which is prime! Coincidentally, 1234567891010987654321 is also prime. If you continue the pattern… it turns out that the next (probable!) prime is the 17350-digit number 1234567…244524462445…7654321 according to OEIS.org.
p=282,589,933 − 1
As we mentioned above in the entry for p=2, if q=mn-1 is a prime number, for some natural number m, then either q=2 or m=2. Moreover, if q=2n-1 is prime, then n is prime (and if so, q is called a Mersenne prime). Unfortunately, this is not a necessary and sufficient criterion and some prime values of n do not yield a Mersenne number q (for instance, 211-1 = 23*89 is composite). The largest known prime (as of the writing of this post) is a Mersenne prime (the 51st Mersenne prime that we have been able to find), namely the prime number M51 = 282,589,933 − 1. It is worth noting the mind-blowing fact that M51 has 24,862,048 digits.
A really cool fact about Mersenne primes is their relationship to even perfect numbers: if 2p-1 is prime, then 2p-1(2p-1) is a perfect number (proved by Euclid!) and, viceversa, if n is an even perfect number, then it is of this form (proved by Euler!). So the largest even perfect number we are aware of is 282,589,932 * (282,589,933 − 1) … a perfect number with 49,724,095 digits!