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
A057002 | Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime). |
 |
Figure 2: cluster of generalised Fermat primes |
No comments:
Post a Comment