Greatest Common Factor
... • So, they are numbers that have more multiples than just one and itself. • For example, the factors of 28 are: • 1 x 28 • 2 x 14 • 4x7 ...
... • So, they are numbers that have more multiples than just one and itself. • For example, the factors of 28 are: • 1 x 28 • 2 x 14 • 4x7 ...
Prime
... This leads to a related theorem… Theorem: If n is a composite integer, then n has a prime divisor less than √n. Proof: If n is composite, then it has a positive integer factor a with 1 < a < n by definition. This means that n = ab, where b is an integer greater than 1. Assume a > √n and b > √n. ...
... This leads to a related theorem… Theorem: If n is a composite integer, then n has a prime divisor less than √n. Proof: If n is composite, then it has a positive integer factor a with 1 < a < n by definition. This means that n = ab, where b is an integer greater than 1. Assume a > √n and b > √n. ...
Number Theory III: Mersenne and Fermat Type Numbers
... I by Professor Don Gillies with the ILLIAC II computer. It was the largest known prime at the time. The discovery was considered sufficiently significant that a special postmark (shown below) was created to celebrate the event. ...
... I by Professor Don Gillies with the ILLIAC II computer. It was the largest known prime at the time. The discovery was considered sufficiently significant that a special postmark (shown below) was created to celebrate the event. ...
prime numbers, complex functions, energy levels and Riemann.
... puzzled people. To understand how the primes are distributed Gauss studied the number (x) of primes less than a given number x. Gauss fund empirically that (x) is approximately given by x/log(x). In 1859 Riemann published a short paper where he established an exact expression for (x). However, th ...
... puzzled people. To understand how the primes are distributed Gauss studied the number (x) of primes less than a given number x. Gauss fund empirically that (x) is approximately given by x/log(x). In 1859 Riemann published a short paper where he established an exact expression for (x). However, th ...
Prime numbers- factor tree File
... We can't factor any more, so we have found the prime factors. Which reveals that 48 = 2 × 2 × 2 × 2 × 3 (or 48 = 24 × 3 using exponents) ...
... We can't factor any more, so we have found the prime factors. Which reveals that 48 = 2 × 2 × 2 × 2 × 3 (or 48 = 24 × 3 using exponents) ...
Prime number theorem
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).The first such distribution found is π(N) ~ N / log(N), where π(N) is the prime-counting function and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N).