I. Precisely complete the following definitions: 1. A natural number n
... 1. The power you want is the minimum of the numbers m and n. Prove this power of p divides a + b by a direct argument. Show no larger power of p divides a + b by contradiction. 2. See the proof by contradiction for the theorem proved in class: there are infinitely many primes of the form 4k + 3. The ...
... 1. The power you want is the minimum of the numbers m and n. Prove this power of p divides a + b by a direct argument. Show no larger power of p divides a + b by contradiction. 2. See the proof by contradiction for the theorem proved in class: there are infinitely many primes of the form 4k + 3. The ...
Full text
... 3511 are the only known such primes. Similarly defined is a Wall-Sun-Sun prime, which is any prime p such that Fp−(5/p) ≡ 0 (mod p2 ), where (5/p) is the Legendre symbol. There are no known Wall-Sun-Sun primes. More generally, in 1993 P. Montgomery added 23 new solutions to ap−1 ≡ 1 (mod p2 ). This ...
... 3511 are the only known such primes. Similarly defined is a Wall-Sun-Sun prime, which is any prime p such that Fp−(5/p) ≡ 0 (mod p2 ), where (5/p) is the Legendre symbol. There are no known Wall-Sun-Sun primes. More generally, in 1993 P. Montgomery added 23 new solutions to ap−1 ≡ 1 (mod p2 ). This ...
HOMEWORK 4: SOLUTIONS - MATH 110 INSTRUCTOR: George
... Suppose that 5 is rational. Then there exist natural numbers m, n, such that 5 = m n. Without loss of generality we may assume that m, n do not have any prime factors in common; otherwise we would have simplified the fraction fraction ...
... Suppose that 5 is rational. Then there exist natural numbers m, n, such that 5 = m n. Without loss of generality we may assume that m, n do not have any prime factors in common; otherwise we would have simplified the fraction fraction ...
Complexité avancée
... Show that BPL ⊆ P . Exercise 2: BPP and oracle machines Prove that PBPP = BPP. Exercise 3: Primality Although the problem PRIMES is now known to be in P (cf AKS Algorithm), the most effective (in practice) primality tests are probabilistic algorithms. In this exercise we analyze one of these probabi ...
... Show that BPL ⊆ P . Exercise 2: BPP and oracle machines Prove that PBPP = BPP. Exercise 3: Primality Although the problem PRIMES is now known to be in P (cf AKS Algorithm), the most effective (in practice) primality tests are probabilistic algorithms. In this exercise we analyze one of these probabi ...
