Primipedia | Obscure Prime Forms

Hi readers! It has been SUCH a long time since we last met! Today we’re going to welcome you back to the second-last article in the Primipedia series. :D

The two primes forms discussed today will be Wieferich and Wall-Sun-Sun primes. These are very special prime forms as well as being very rare. We』ll also have a very special announcement for you!

If you don’t want to read the mathematical descriptions and just want the special announcement, please jump right to the end!

1. Wieferich Primes

Firstly we need to remind you that you DO NOT need to remember these primes』 forms. They’re complicated. Let’s take a look:

A prime is a Wieferich prime if 

This sounds complicated, right? Well, our dear Bob has an answer for us. Bob says, 「These primes are actually a special case of Fermat’s Little Theorem.」

But why?

Let’s take a look back at Fermat’s Little Theorem that we』ve covered previously: A number  divides  (or in general,

Let’s take

So who had this idea of a 『special case』 prime? These primes are named after Arthur Wieferich who in 1909 proved that if a prime 

Despite a number of extensive searches, the only known Wieferich primes to date are 1093 and 3511. The rarity of these primes has lead to an interest in "Near" Wieferich primes. They are defined as special instances:

A prime 

Let’s see an example of a Wieferich and an example of a near-Wieferich with

3511 is the largest known Wieferich prime to date (Nov. 8th, 2020), so

We can check this statement’s credulity by using pfgw (a software). We move the 1 to the left side of the equation to get

the largest known with

Firstly we have

Using the WWocl application (all mentioned applications will be given at the end), we are able to determine the above relation to be true. Therefore we have the result that

2. Wall-Sun-Sun Primes

Wall-Sun-Sun primes are also called Fibonacci-Wieferich primes, and they have even more complicated forms.They’re defined as primes

Again, you question the meaning of such primes. And, yes, you expected it, Bob comes to answer your problem. He says, 「Like Wieferich primes, Wall-Sun-Sun primes are rejects from the Fermat’s Last Theorem. Primes that don’t satisfy the first case of the aforementioned theorem are all Wall-Sun-Sun primes.」

At this, you relate this with the fact that twin brothers Zhihong Sun and Zhiwei Sun, drawing on Donald Dines Wall’s work, proved the above statement in 1992. Although infinite such primes are conjectured to exist, none are known yet. This lack of success has lead to an interest in Near-Wall-Sun-Sun primes.

These are in the form of

And that concludes our prime journey today.

PECIA

Wait.

WHERE ARE THE PROMISED SPECIAL ANNOUNCEMENTS???

Ah yes.

On 25 October 2020, 11:30:07 UTC, the Thabit prime was discovered by Dr. James Scott Brown. The prime is 4,939,547 digits long and enters Chris Caldwell's 「The Largest Known Primes Database」 ranked 21st overall.

And: THE WW SEARCH IS COMING TO PRIMEGRID! Have you wondered why we’re talking about such complicated prime forms? That’s because PrimeGrid is launching a search for these two prime forms! If this search succeeds, we』ll be the discoverers of the third Wieferich prime and the very first Wall-Sun-Sun prime!

Testing software is available at:https://github.com/mfl0p/WWcpu-boinc/tree/master/bin

Thank you for reading, and tune in for our last episode! Hope you liked it!

Written by Daniel Lv
Designed by Winola Deng