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 ...
slides - faculty.ucmerced.edu
... • For example, 2 comparisons are used when the list has 2k-1 elements, 2 comparisons are used when the list has 2k-2, …, 2 comparisons are used when the list has 21 elements • 1 comparison is ued when the list has 1 element, and 1 more comparison is used to determine this term is x • Hence, at most ...
... • For example, 2 comparisons are used when the list has 2k-1 elements, 2 comparisons are used when the list has 2k-2, …, 2 comparisons are used when the list has 21 elements • 1 comparison is ued when the list has 1 element, and 1 more comparison is used to determine this term is x • Hence, at most ...
On the greatest prime factor of sides of a Heron triangle
... in Z[i ] also. If p ≡ 1 (mod 4), then p = a 2 + b2 = (a + bi )(a − bi ) and both numbers a + bi and a − bi are primes in Z[i ]. Furthermore, 2 = i (1 − i )2 and 1 − i is a prime. Finally, all primes in Z[i ] are obtained as above up to multiplication by units. Let (a, b, c) be the sides of one of th ...
... in Z[i ] also. If p ≡ 1 (mod 4), then p = a 2 + b2 = (a + bi )(a − bi ) and both numbers a + bi and a − bi are primes in Z[i ]. Furthermore, 2 = i (1 − i )2 and 1 − i is a prime. Finally, all primes in Z[i ] are obtained as above up to multiplication by units. Let (a, b, c) be the sides of one of th ...
Adding and Subtracting Fractions Guided Notes
... I can ____________ and ________________ ____________________________. Adding and Subtracting Fractions Vocabulary Numerator - the number on ____________ in a fraction. Denominator - the number on ______________________ in a fraction. Least Common Multiple - the _________________ number that is in __ ...
... I can ____________ and ________________ ____________________________. Adding and Subtracting Fractions Vocabulary Numerator - the number on ____________ in a fraction. Denominator - the number on ______________________ in a fraction. Least Common Multiple - the _________________ number that is in __ ...
number_theory_handout_II
... 3. How many even positive divisors does 1000 have? What is their sum? 4. What is the product of all positive divisors of 500? 5. If n = pk11 . . . pkmm is the prime factorization of n, find a formula for the product of all positive divisors of n. 6. Calculate 2123 (mod 11). 7. Calculate 2123 + 5123 ...
... 3. How many even positive divisors does 1000 have? What is their sum? 4. What is the product of all positive divisors of 500? 5. If n = pk11 . . . pkmm is the prime factorization of n, find a formula for the product of all positive divisors of n. 6. Calculate 2123 (mod 11). 7. Calculate 2123 + 5123 ...
Sieve of Eratosthenes
In mathematics, the sieve of Eratosthenes (Ancient Greek: κόσκινον Ἐρατοσθένους, kóskinon Eratosthénous), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the multiples of 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. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.The sieve of Eratosthenes is one of the most efficient ways to find all of the smaller primes. It is named after Eratosthenes of Cyrene, a Greek mathematician; although none of his works have survived, the sieve was described and attributed to Eratosthenes in the Introduction to Arithmetic by Nicomachus.The sieve may be used to find primes in arithmetic progressions.