Homework Answers
Let p be a prime. Consider the sequence 11,22,3, 44,55 modulo p. Prove that the resulting...
Let p be a prime. Consider the sequence 11,22,3, 44,55 modulo p. Prove that the resulting sequence is periodic with smallest period p(p - 1). (This means that p(p - 1) is the least among all positive integers l with the property that whenever n = m (mod l), we have n" = m" (mod p).)
Let p be a prime. Consider the sequence 11,22,3, 44,55 modulo p. Prove that the resulting sequence is periodic with smallest period p(p -...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × ...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)]
27. (a) Let...
In the power method, let rk d(x(k+1))/ф(z(k)). We know that limk-oo rk Show that the relative err...
Please answer both part. Thanks.
In the power method, let rk d(x(k+1))/ф(z(k)). We know that limk-oo rk Show that the relative errors obey 1- Ai where the numbers ck form a convergent (and hence bounded) sequence. (Continuation) Show that rk +1-λι-(c+&J(rk-A) where Icl < 1 and limn-o0 Sk 0, so that Aitken acceleration is applicable.
In the power method, let rk d(x(k+1))/ф(z(k)). We know that limk-oo rk Show that the relative errors obey 1- Ai where the numbers ck form...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,...,...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
8.15 Lemma. Let p be a prime and let a be a natural mumber not divisible by p. Then there exist integers x and y such t...
8.15 Lemma. Let p be a prime and let a be a natural mumber not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lylP 8.16 Theorem. Ler p be a prime such that p 1 (mod 4). Then p is equal to the sum of two squares of natural numbers. (int: Iry applying the previous lemma to a square root of- mochdo p.) Knowing which primes can be written as the...
Let J be the ring of integers, p a prime number, and () the ideal of...
Let J be the ring of integers, p a prime number, and () the ideal of J consisting of all multiples of p. Prove (a) J@) is isomorphic to Jp, the ring of integers mod p. (b) Using Theorem 3.5.1 and part (a) of this problem, that J, is a field.
Let k and a be two positive integers, such that ak-1 = 1(mod k) and gcd(k, a) = 1. Is k prime or composite? If so why and explain all the steps. Thanks
Let k and a be two positive integers, such that ak-1 = 1(mod k) and gcd(k, a) = 1. Is k prime or composite? If so why and explain all the steps. Thanks
2. Let p be an odd prime. We saw last week that the problem of counting solutions to the congruen...
2. Let p be an odd prime. We saw last week that the problem of counting solutions to the congruence (mod p) is only interesting when p has the form 4k1. For the rest of this problem let p 4k+1. (a) Show that (mod -1 5 (mod 8) (b) Show that pEl (mod 8 -1 p5 (mod 8) (c) Draw condlusions about the number of solutions to these congruences 14 (mod p) -1 (mod p) (mod p)
2. Let p...
Let p be an odd prime. Write p in the form p = 2k + 1...
Let p be an odd prime. Write p in the form p = 2k + 1 for some k E N. Prove that kl-(-1)* mod p. Hint: Each j e Z satisfies j (p-od p.
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers....
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
Let p be a prime. Consider the sequence 11,22,3, 44,55 modulo p. Prove that the resulting sequence is periodic with smallest period p(p - 1). (This means that p(p - 1) is the least among all positive integers l with the property that whenever n = m (mod l), we have n" = m" (mod p).)
Let p be a prime. Consider the sequence 11,22,3, 44,55 modulo p. Prove that the resulting sequence is periodic with smallest period p(p -...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)]
27. (a) Let...
Please answer both part. Thanks.
In the power method, let rk d(x(k+1))/ф(z(k)). We know that limk-oo rk Show that the relative errors obey 1- Ai where the numbers ck form a convergent (and hence bounded) sequence. (Continuation) Show that rk +1-λι-(c+&J(rk-A) where Icl < 1 and limn-o0 Sk 0, so that Aitken acceleration is applicable.
In the power method, let rk d(x(k+1))/ф(z(k)). We know that limk-oo rk Show that the relative errors obey 1- Ai where the numbers ck form...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
8.15 Lemma. Let p be a prime and let a be a natural mumber not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lylP 8.16 Theorem. Ler p be a prime such that p 1 (mod 4). Then p is equal to the sum of two squares of natural numbers. (int: Iry applying the previous lemma to a square root of- mochdo p.) Knowing which primes can be written as the...
Let J be the ring of integers, p a prime number, and () the ideal of J consisting of all multiples of p. Prove (a) J@) is isomorphic to Jp, the ring of integers mod p. (b) Using Theorem 3.5.1 and part (a) of this problem, that J, is a field.
2. Let p be an odd prime. We saw last week that the problem of counting solutions to the congruence (mod p) is only interesting when p has the form 4k1. For the rest of this problem let p 4k+1. (a) Show that (mod -1 5 (mod 8) (b) Show that pEl (mod 8 -1 p5 (mod 8) (c) Draw condlusions about the number of solutions to these congruences 14 (mod p) -1 (mod p) (mod p)
2. Let p...
Let p be an odd prime. Write p in the form p = 2k + 1 for some k E N. Prove that kl-(-1)* mod p. Hint: Each j e Z satisfies j (p-od p.
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
Most questions answered within 3 hours.
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A. 0≤P(Oi)≤10≤P(Oi)≤1 for each i
B. P(Oi)≤0P(Oi)≤0
C. P(Oi)=1+P(OCi)P(Oi)=1+P(OiC)
D. P(Oi)≥1P(Oi)≥1
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