Fermat number

Fermat prime
Named afterPierre de Fermat
No. of known terms5
Conjectured no. of terms5
Subsequence ofFermat numbers
First terms3, 5, 17, 257, 65537
Largest known term65537

In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form

where n is a nonnegative integer. The first few Fermat numbers are:

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … (sequence A000215 in the OEIS).

If 2k + 1 is prime, and k > 0, it can be shown that k must be a power of two. (If k = ab where 1 ≤ a, bk and b is odd, then 2k + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). See below for a complete proof.) In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1) is a Fermat number, and such primes are called Fermat primes. As of 2019, the only known Fermat primes are F0, F1, F2, F3, and F4 (sequence A019434 in the OEIS).

Basic properties

The Fermat numbers satisfy the following recurrence relations:

for n ≥ 1,

for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both

and Fj; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each Fn, choose a prime factor pn; then the sequence {pn} is an infinite sequence of distinct primes.

Further properties:

Other Languages
Ænglisc: Fermat tæl
العربية: عدد فيرما
azərbaycanca: Ferma ədədləri
български: Число на Ферма
Deutsch: Fermat-Zahl
Ελληνικά: Αριθμός Φερμά
Esperanto: Nombro de Fermat
français: Nombre de Fermat
한국어: 페르마 수
հայերեն: Ֆերմայի թիվ
עברית: מספר פרמה
Nederlands: Fermatgetal
norsk nynorsk: Fermattal
Piemontèis: Nùmer ëd Fermat
português: Número de Fermat
русский: Число Ферма
Simple English: Fermat number
slovenščina: Fermatovo praštevilo
svenska: Fermattal
українська: Числа Ферма
Tiếng Việt: Số Fermat
中文: 費馬數