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Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work Elements . There are several proofs of the theorem.
Not all Euclid numbers are prime. E 6 = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number.. Every Euclid number is congruent to 3 modulo 4 since the primorial of which it is composed is twice the product of only odd primes and thus congruent to 2 modulo 4.
The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2 p −1 (2 p − 1) , where 2 p − 1 is a prime number .
Any prime number is prime to any number it does not measure. [note 7] Proposition 30 If two numbers, by multiplying one another, make the same number, and any prime number measures the product, it also measures one of the original numbers. [note 8] Proof of 30 If c, a prime number, measure ab, c measures either a or b. Suppose c does not measure a.
Many more proofs of the infinitude of primes are known, including an analytical proof by Euler, Goldbach's proof based on Fermat numbers, [52] Furstenberg's proof using general topology, [53] and Kummer's elegant proof. [54] Euclid's proof [55] shows that every finite list of primes is incomplete.
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements. If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. —
A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem , there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes .
As of December 2024, the largest known prime of the form p n # + 1 is 7351117# + 1 (n = 498,865) with 3,191,401 digits, also found by the PrimeGrid project. Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner: [2]