Question

Come up with a rule for when 5 is a quadratic residue mod p using quadratic reciprocity.

Answer #1

Let p be an odd prime.
Prove that −1 is a quadratic residue modulo p if p ≡ 1 (mod 4),
and −1 is a quadratic nonresidue modulo p if p ≡ 3 (mod 4).

Let p be be prime and p ≡ 1 (mod 4|a|). Prove that a is a
quadratic residue mod p.

Consider the above quadratic residue generator xn+1 =
xn2 mod m with m = 4783 × 4027. Write a
program to generate pseudo-random numbers from this generator. Use
this to determine the period of the generator starting with seed
x0 = 196, and with seed x0 = 400?

Compute the remainder of:
5^(31) mod 1537 and 19^(31) mod 1537 USING a combination of
euler's theorem and fermats little theorem

COME UP WITH A PROBLEM FOR YOUR CLASSMATES TO SOLVE INVOLVING
THE ADDITION RULE, MULTIPLICATION RULE, PERMUTATIONS (WITH OR
WITHOUT REPLACEMENT), OR COMBINATIONS (WITH OR WITHOUT
REPLACEMENT). HOWEVER, THE PROBLEM YOU POST IS ALSO ONE OF YOUR
HOMEWORK PROBLEMS AND YOU MUST GIVE ITS PROBLEM TYPE AND A CORRECT
SOLUTION THERE. FOR "REVIEWS" SOLVE THE PROBLEMS OF TWO (2)
CLASSMATES FIRST TELLING WHICH TYPE PROBLEM IT IS (E.G., ADDITION
RULE), THEN SHOWING YOUR WORK WITH YOUR ANSWER (CORRECT ANSWERS TO...

Prove Fermat’s Little Theorem using induction: ap ≡ a (mod p)
for any a ∈Z.

Come up with a 2 x 2 contingency table for which p^ 1=p^ 2 , and
for which the row and column totals are not the same. Now calculate
χ 2*. Are you surprised? Why or why not? (pls include the
table)

Perform encryption using the RSA algorithm for the
following:
p=5, q=9, e=2, M=5
p=4, q=12, e=4, M=3
C=Me mod n
C is the cipher text and M is the plain text, n=p×q

Draw a 4-bit mod-8 counting up asynchronous ripple counter with
self-stopping function using JK FF(s) and suitable logic
gate(s).

What is the rejection rule using the p-value approach to
Hypothesis Testing?
What is the rejection rule using the critical value approach to
Hypothesis Testing?

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