PPT
... • An integer p is prime if its only divisors are 1 and p • An integer that is greater than 1, and not prime is called composite • Fundamental theorem of arithmetic: – Every positive integer greater than one has a unique prime factorization ...
... • An integer p is prime if its only divisors are 1 and p • An integer that is greater than 1, and not prime is called composite • Fundamental theorem of arithmetic: – Every positive integer greater than one has a unique prime factorization ...
Solutions
... 13. What is the multiplicity of the prime factors of 9 765 625? The prime facotorization is 9 765 625 = 59 . The multiplicity of 5 is 9. 14. Is 129 a Mersenne prime? No, the closest Mersenne prime to 129 is 127 = 27 − 1. 15. * Locker Problem Extended Which locker(s), from 1 to 100, were open and cl ...
... 13. What is the multiplicity of the prime factors of 9 765 625? The prime facotorization is 9 765 625 = 59 . The multiplicity of 5 is 9. 14. Is 129 a Mersenne prime? No, the closest Mersenne prime to 129 is 127 = 27 − 1. 15. * Locker Problem Extended Which locker(s), from 1 to 100, were open and cl ...
Greatest Common Factor (GCF)
... are smaller it is easiest to list out the factors of each number and look for the biggest one that they have in common. 1. List the factors of each number. 2. Find the greatest factor that they have in common. EXAMPLE: Find the GCF of 24 and 30. 24: The factors are 1, 2, 3, 4, 6, 8, 12, 24 30: The f ...
... are smaller it is easiest to list out the factors of each number and look for the biggest one that they have in common. 1. List the factors of each number. 2. Find the greatest factor that they have in common. EXAMPLE: Find the GCF of 24 and 30. 24: The factors are 1, 2, 3, 4, 6, 8, 12, 24 30: The f ...
Real Numbers
... Euclid’s division lemma/algorithm has several applications related to finding properties of numbers. We give some examples of these applications below: Example 2 : Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some inte ...
... Euclid’s division lemma/algorithm has several applications related to finding properties of numbers. We give some examples of these applications below: Example 2 : Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some inte ...
Document
... powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain. ...
... powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain. ...
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).