Generalised Fermat Primes

Today I turned 26188 days old and this number happens to be associated with the so-called generalised Fermat primes. Before going further, we should establish what is meant by a Fermat prime and a Fermat number. 

A Fermat number $F_n$ is a number such that $F_n=2^{2^n}+1$. 

If the number is prime, then we have a Fermat prime. Currently, only five such primes are known and these are:

 $F_0=3, F_1=5, F_2=17, F_3=257, F_4=65537$

A generalised Fermat number is a number of the form $a^{2^n}+1$ where $a>2$. Only if $a$ is even can a generalised Fermat number be prime. Now 1024=$2^{10}$ and so if we look at numbers of the form $a^{2^{10}}+1$, we find that the values of $a$ that produce primes are (up to 26188): 

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188

These numbers form OEIS A057002. Figure 1 shows a plot of these same numbers:

Figure 1

Many of the largest known prime numbers are generalised Fermat numbers. To date (14th December 2020), the largest such prime is:

$$1059094^{2^{20}}+1=1059094^{1048576}+1 \text{ which contains } 6317602 \text{ digits }$$ This prime was discovered in 2018 but it pales in comparison to a number of larger Mersenne primes, the largest of which (discovered also in 2018) is: $$2^{82589933-1} \text{ which contains } 24862048 \text{ digits }$$ A list of the current largest 100 primes can be found here.

It should be noted that there is another less common definition of a generalised Fermat number and that is:

$$F_m(a,b)=a^{2m}+b^{2m} \text{ with gcd(\(a,b\))=1}$$ I've looked at generalisations or extensions of other number types in the past, specifically:

• Generalised Cunningham Chains
• Fibonacci-like Sequences
• Beyond Fibonacci
• Root-Mean-Square and other Means

UPDATE on Thursday, February 4th 2021

Today I turned 26240 days old and one of the properties of this number, as with 26188 that is dealt with in this post, is that it is a generalised Fermat prime. Furthermore, both numbers belong to OEIS A057002:

A057002

Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

In fact, looking at the members of the sequence, we see that there is a cluster of three numbers (26132, 26188, 26240) and Figure 2 makes this even more apparent:

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016, ...

Figure 2: cluster of generalised Fermat primes