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A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.
It follows from this fact that for every prime p > 2, there is at least one prime of the form 2kp+1 less than or equal to M p, for some integer k. If p is an odd prime, then every prime q that divides 2 p − 1 is congruent to ±1 (mod 8). Proof: 2 p+1 ≡ 2 (mod q), so 2 1 / 2 (p+1) is a square root of 2 mod q.
Prime gap probability density for primes up to 1 million. Peaks occur at multiples of 6. [1]A prime gap is the difference between two successive prime numbers.The n-th prime gap, denoted g n or g(p n) is the difference between the (n + 1)-st and the n-th prime numbers, i.e.
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
If one wishes to prove a statement, not for all natural numbers, but only for all numbers n greater than or equal to a certain number b, then the proof by induction consists of the following: Showing that the statement holds when n = b. Showing that if the statement holds for an arbitrary number n ≥ b, then the same statement also holds for n ...
A Wieferich number is an odd natural number n satisfying the congruence 2 φ (n) ≡ 1 (mod n 2), where φ denotes the Euler's totient function (according to Euler's theorem, 2 φ (n) ≡ 1 (mod n) for every odd natural number n). If Wieferich number n is prime, then it is a Wieferich prime. The first few Wieferich numbers are:
The Pólya conjecture states that for any n > 1, if the natural numbers less than or equal to n (excluding 0) are partitioned into those with an odd number of prime factors and those with an even number of prime factors, then the former set has at least as many members as the latter set. Repeated prime factors are counted repeatedly; for ...