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Problem 2: For each of the following numbers a find the set ...

Problem 2: For each of the following numbers a find the set of primes p such that a is a quadratic residue modulo p: a) (1 pts) 10. b) (1 pts) - -3. Problem 3: (6 pts) Let p be an odd prime and a, b E (Z/pZ)*. Then, show that the number of solutions to ax² + by² = 1 mod p is p - (-A...

I. Find the following (Show your complete solution.) 1. The...

I. Find the following (Show your complete solution.) 1. The remainder when 5¹⁴⁰ is divided by 11. [15 points] 2. All positive integers n for which 11 ≡ 4 (mod n). [10 points] 3. The general solution to 15x ≡ 11 (mod 73). [10 points 4. All integers between 1000 and 2000 that ...

1. Find the number of positive integers not exceeding 1000 t...

1. Find the number of positive integers not exceeding 1000 that are not divisible by 3,3 or 7. 2. Find all pairs of integers a and b such that (a,b) = 18 and [a.b] = 540. 3. Find us and U so that 272u + 1479c = d, where d is the greatest common divisor of 272 and 1479. 4. Find 8 solution: 803x + ...

3 Analysis Give a description of each algorithm which is su...

3 Analysis Give a description of each algorithm which is sufficiently detailed to allow you to give an estimate of the average and worst case running time (as a function of k) for each of your chosen algorithms on the set K. Give your analysis, and describe numerically what the distribution of runn...

Algebra 1. (a) State the definition of agroup. (b) State w...

Algebra 1. (a) State the definition of agroup. (b) State without proof whether or not the following are examples of groups i. The set Q' of non-zero rational numbers under multiplication ii. The set of symmetries of a cube under composition. iii. The set {-1,0,+1} under addition. (c) Prove tha...

(a) Consider the Lagrangian n n L = [ - 4j + (g) t2 1 [ a...

(a) Consider the Lagrangian n n L = [ - 4j + (g) t2 1 [ ajkqjik, j=1 where Ej. ajk, j,k = 1, n are constants and ajk = -akj. We assume that q;>0,j=1, = n. Find the Euler-Lagrange equations and write them in the form 4j = Fi(q,q). (b) Compute the canonically conjugate variables Pj and ...

(a) Consider the Lagrangian n n 1 L = - + (q);) +2 j=...

(a) Consider the Lagrangian n n 1 L = - + (q);) +2 j=1 j,k=1 where Ej, ajk j,k = 1, , 1 n are constants and ajk = -akj We assume that 4j > 0, j = 1,...,n. Find the Euler-Lagrange equations and write them in the form Äj = Fj(q,q). (b) Compute the canonically conjugate variable...

(a) Consider the Lagrangian n n 1 L = [ (àj In 9...

(a) Consider the Lagrangian n n 1 L = [ (àj In 9j - 9j + (q);) + - ajk4jqk 2 j=1 j,k=1 where Ej, ajk, jk = 1, , n are constants and ajk = -akj. We assume that qj > 0, j = 1 , n. , Find the Euler-Lagrange equations and write them in the form 9j = Fi(q,q).

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