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this is number theory i need help with thanks alsonlls show
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Assume a, b,...are...
Need help!! Please help — crypto math 1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root...
Need help!! Please help — crypto math
1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root for the prime 1223. You want to solve the discrete logarithm problem 53 (mod 1223). You know 3611 Prove it. 1 (mod 1223). Is x even or odd?
1. Determine L13(18) for p 19. 2. Let p be prime,...
I need help with number 3 on my number theory hw. Exercise 1. Figure out how...
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrut...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive...
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive integers n such that n|10! and gcd(n, 27.34.7) = 27.3.7. Justify your answer. Question 4 Let m, n E N. Prove that ged(m2, n2) = (gcd(m, n))2. Question 5 Let p and q be consecutive odd primes with p < q. Prove that (p + q) has at least three prime divisors (not necessarily distinct).
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b...
please prove proofs and do
7.4
7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
Number theory: Part C and Part D please! QUADRA range's Four-Square Theorem) If n is a natural be expressed as t...
Number theory: Part C and Part D please!
QUADRA range's Four-Square Theorem) If n is a natural be expressed as the sum of four squares. insmber, then n cam be expressed tice Λ in 4-space is a set of the form t(x,y, z, w). M:x,y,z, w Z) matrix of nonzero determinant. The covolume re M is a 4-by-4 no is defined to be the absolute value of Det M such a lattice, of covolume V, and let S be the...
please prove lemma and theorems. 8.17 is not needed, thank you 8.15 Lemma. Let p be...
please prove lemma and
theorems. 8.17 is not needed, thank you
8.15 Lemma. Let p be a prime and let a be a natural number not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lyl 8.16 Theorem. Let p be a prime such that p (mod 4). Thenp is equal to the sum of two squares of natural numbers. (Hinl: Iry applying the previous lemma to a square root of- mohulo...
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...
solve it ,i need urgent, no need to write neat and clean.. thanks! ......b0nGrr....... 1. (a)...
solve it ,i need urgent, no need to write neat and clean..
thanks!
......b0nGrr.......
1. (a) Find real numbers a and b such that a + bi = p.v.(-86]1/3. [4] (b) Consider the following statement. "Log(-x)2 = Logza because (-2)2 = 22. Therefore, 2 Log(-2) = 2 Logz." Explain whether or not the statement is true. [4] (c) Consider the following statement. "The rational function (2), where p and q are co-prime non-constant polynomials, is holomorphic everywhere except at the...
We know that we can reduce the base of an exponent modulo m: a(a mod m)k...
We know that we can reduce the base of an exponent modulo m: a(a mod m)k (mod m). But the same is not true of the exponent itself! That is, we cannot write aa mod m (mod m). This is easily seen to be false in general. Consider, for instance, that 210 mod 3 1 but 210 mod 3 mod 3 21 mod 3-2. The correct law for the exponent is more subtle. We will prove it in steps (a)...
Need help!! Please help — crypto math
1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root for the prime 1223. You want to solve the discrete logarithm problem 53 (mod 1223). You know 3611 Prove it. 1 (mod 1223). Is x even or odd?
1. Determine L13(18) for p 19. 2. Let p be prime,...
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive integers n such that n|10! and gcd(n, 27.34.7) = 27.3.7. Justify your answer. Question 4 Let m, n E N. Prove that ged(m2, n2) = (gcd(m, n))2. Question 5 Let p and q be consecutive odd primes with p < q. Prove that (p + q) has at least three prime divisors (not necessarily distinct).
please prove proofs and do
7.4
7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
Number theory: Part C and Part D please!
QUADRA range's Four-Square Theorem) If n is a natural be expressed as the sum of four squares. insmber, then n cam be expressed tice Λ in 4-space is a set of the form t(x,y, z, w). M:x,y,z, w Z) matrix of nonzero determinant. The covolume re M is a 4-by-4 no is defined to be the absolute value of Det M such a lattice, of covolume V, and let S be the...
please prove lemma and
theorems. 8.17 is not needed, thank you
8.15 Lemma. Let p be a prime and let a be a natural number not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lyl 8.16 Theorem. Let p be a prime such that p (mod 4). Thenp is equal to the sum of two squares of natural numbers. (Hinl: Iry applying the previous lemma to a square root of- mohulo...
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...
solve it ,i need urgent, no need to write neat and clean..
thanks!
......b0nGrr.......
1. (a) Find real numbers a and b such that a + bi = p.v.(-86]1/3. [4] (b) Consider the following statement. "Log(-x)2 = Logza because (-2)2 = 22. Therefore, 2 Log(-2) = 2 Logz." Explain whether or not the statement is true. [4] (c) Consider the following statement. "The rational function (2), where p and q are co-prime non-constant polynomials, is holomorphic everywhere except at the...
We know that we can reduce the base of an exponent modulo m: a(a mod m)k (mod m). But the same is not true of the exponent itself! That is, we cannot write aa mod m (mod m). This is easily seen to be false in general. Consider, for instance, that 210 mod 3 1 but 210 mod 3 mod 3 21 mod 3-2. The correct law for the exponent is more subtle. We will prove it in steps (a)...
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