Number theory, modular arithmetic
Prove that is p is prime, we have: n choose m is congruent to [floor(n/p) choose floor(m/p)]*[(n mod p) choose (m mod p)] (mod p) Hint: show that (1+x)^(pq+r) is congruent to (1+x)^r * (1+x^p)^q (mod p) If you can point me to a book or website explaining how to do this type of problem, and give a sketch of the proof, ...continues
Show that log(r), where this is log base 10, is irrational when r is a positive rational that is not an integral power of 10 I have already proven that e^r is irrational for all rational numbers r
Let m, n be in N, with m, n >= 1 and n odd. Let S_m = 1^n + 2^n + 3^n + ... + m^n. Prove that S_m is divisible by 1+2+...+m.
Let b = r_0, r_1, r_2, ... be the successive remainders in the Euclidean Algorithm applied to a and b. Show that every 2 steps reduces the remainder by at least one half. In other words, verify that r_{i+2} < (1/2)r_{i}, for every i = 0,1,2,.... Conclude that the Euclidean algorithm terminates in at most 2log_{2}(b) steps, where ...continues
Explain what is meant by a linear code over Fq, the weight of a vector and the weight of a linear code. See attached file for full problem description.
Let C be the linear -code with generator matrix Prove that C is a perfect code. See attached file for full problem description.
Coding theory - cyclic codes. See attached file for full problem description.
Let x+C be a coset, and assume x+c, x+e have weight less or equal than t. See attached file for full problem description.
Coding theory - sphere packing
Coding theory - sphere packing. See attached file for full problem description.
Polynomials (specifically palindromic) with Z mod coefficients
I need to do some research on the properties of palindromic polynomials with Z(n) coefficients. I would like information/explanation of polynomials with Z(n) coefficients. I would like to see examples of polynomials with Z(1), Z(2), Z(3), Z(4), Z(5) and in general Z(n) coefficients. Also, I would like to see some examples of ...continues
