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Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between and (+) for every positive integer. [1] The conjecture is one of Landau's problems (1912) on prime numbers, and is one of many open problems on the spacing of prime numbers.
Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime? Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares? Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n 2 + 1? As of 2025, all ...
In 1737, Euler related the study of prime numbers to what is known now as the Riemann zeta function: he showed that the value () reduces to a ratio of two infinite products, Π p / Π (p–1), for all primes p, and that the ratio is infinite. [1] [2] In 1775, Euler stated the theorem for the cases of a + nd, where a = 1. [3]
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime. [1] This is the sieve's key distinction from ...
The prime number theorem asserts that an integer m selected at random has roughly a 1 / ln m chance of being prime. Thus if n is a large even integer and m is a number between 3 and n / 2 , then one might expect the probability of m and n − m simultaneously being prime to be 1 / ln m ln(n − m) .
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If it is 1, then n may be prime. If a n −1 (modulo n) is 1 but n is not prime, then n is called a pseudoprime to base a. In practice, if a n −1 (modulo n) is 1, then n is usually prime. But here is a counterexample: if n = 341 and a = 2, then even though 341 = 11·31 is composite.
The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is 1/2") and other prime-number problems, among them Goldbach's conjecture and the twin prime conjecture: Unresolved. — 9th: Find the most general law of the reciprocity theorem in any algebraic number field. Partially resolved.