Fermat number

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"text": "Fermat numbers are a specific sequence of positive integers. They are named after Pierre de Fermat, a mathematician who lived from 1601 to 1665. Fermat was the first person known to have studied these numbers. A Fermat number is created using a specific mathematical form. For any non-negative integer $n$, the $n$th Fermat number, denoted as $F_n$, is calculated as $2^{2^n} + 1$. This formula causes the numbers to grow at an incredibly rapid rate. Because they grow so quickly, they present unique challenges for mathematicians.

\n\nTo understand how they work, we can look at the first few values in the sequence. When $n$ is 0, the result is 3. When $n$ is 1, the result is 5. The next values are 17, 257, and 65,537. These first five numbers are all prime numbers. A prime number is an integer greater than 1 that has no divisors other than 1 and itself. When a Fermat number is prime, it is called a Fermat prime. However, the sequence quickly produces much larger and more complex numbers. For example, $F_5$ is 4,294,967,297. As $n$ increases, the number of digits grows extremely fast. $F_{10}$ contains 309 digits, making it very difficult to process.\n\nThese numbers follow several interesting mathematical rules and relations. For example, they satisfy specific recurrence relations. For $n \ge 1$, one relation is $F_n = (F_{n-1} - 1)^2 + 1$. Another relation is $F_n = F_0 \times F_1 \times ... \times F_{n-1} + 2$. These properties allow mathematicians to prove things about the sequence using mathematical induction. One important result is Goldbach's theorem. Named after Christian Goldbach, this theorem states that no two Fermat numbers share a common integer factor greater than 1. This means every Fermat number is coprime to every other Fermat number in the sequence. This fact provides a way to prove that there are infinitely many prime numbers.\n\nHistory shows that Pierre de Fermat originally conjectured that all Fermat numbers were prime. He likely based this on the first five numbers, which are all prime. However, this conjecture was eventually proven wrong. In 1732, Leonhard Euler refuted Fermat's idea. Euler discovered that $F_5$ is actually a composite number, meaning it can be divided by numbers other than 1 and itself. He showed that 641 is a factor of $F_5$. Later, Édouard Lucas improved upon Euler's work. He proved that every factor of a Fermat number $F_n$ (where $n \ge 2$) must be of the form $k \cdot 2^{n+2} + 1$. This helps mathematicians narrow down the search for factors.\n\nToday, many questions about Fermat numbers remain unsolved. Mathematicians do not know if there are infinitely many Fermat primes. There are also open questions about whether there are infinitely many composite Fermat numbers. We do not know if a Fermat number exists that is not square-free. Currently, $F_0, F_1, F_2, F_3,$ and $F_4$ are the only known Fermat primes. Beyond $F_4$, no other Fermat primes have been found. Some researchers use heuristic arguments to suggest that $F_4$ might be the last one. These arguments suggest the probability of finding another Fermat prime is extremely low.

\n\nBecause these numbers are so massive, finding their factors requires immense computing power. The distributed computing project Fermatsearch works to find new factors. We have only completely factored the first twelve Fermat numbers. The largest known composite Fermat number is $F_{18233954}$, and its prime factor was found in October 2020. Tools like the elliptic curve method help find small prime divisors. For primality testing, mathematicians use Pépin's test. This test states that for $n \ge 1$, $F_n$ is prime if and only if $3^{(F_n-1)/2} \equiv -1 \pmod{F_n}$. This is a fast way to check a number, but the size of Fermat numbers still limits how many can be tested.\n\nFermat numbers also have a deep connection to geometry. This connection is explained by the Gauss-Wantzel theorem. This theorem describes which regular polygons can be constructed using only a compass and a straightedge. A regular $n$-sided polygon is constructible if $n$ is the product of a power of 2 and distinct Fermat primes. This means the number of sides must follow the form $n = 2^k \cdot p_1 \cdot p_2 \cdot ... \cdot p_s$. Here, $k$ and $s$ are non-negative integers, and each $p$ is a distinct Fermat prime. This links the abstract world of number theory to the physical world of shapes and constructions.

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"File:Constructible polygon set.svg"

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}