Question

Find all primes p such that p | (a^37−a) for all a ∈N. Multiply
those primes together to ﬁnd the largest n ∈N such that n | (a^37
−a) for all a ∈N.

Answer #1

For each prime number p below, find all of the Gaussian primes q
such that p lies below q:
2
3
5
Then for each Gaussian prime q below, find the prime number p
such that q lies above p:
1 + 4i
3i
2 + 3i

Suppose that n is a product of two k-bit primes p and q. Suppose
also that it is known that
|p-q|<2t, where t is small. DESCRIBE a way to find the
factorization of n in t steps. (Note: in terms of RSA, it shows
that although we want p and q to be of similar size, it is also
undesirable that p and q are very close)

Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

Number Theory:
Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

Find the first 100 primes found by the classical proof of the
infinitude of the set of primes. That is: begin with P={2}; then
form m, the sum of 1 with the product over all elements of P. Place
the smallest prime factor of m into P and repeat. (For this
problem, you may use FactorIntegeror similar built-in Mathematica
functions.). Produce a Mathematica procedure with the above
instructions.

Let phi(n) = integers from 1 to (n-1) that are relatively prime
to n
1. Find phi(2^n)
2. Find phi(p^n)
3. Find phi(p•q) where p, q are distinct primes
4. Find phi(a•b) where a, b are relatively prime

p = 13
q = 37
Totient = 432
n = 481
e = 19
d = 91
public key: n = 481, e = 19
private key: n= 481, d = 91
Except 1, explain any other prime number less than e which
cannot be used to generate d? What are those number and why they
cannot be used?

Consider all integers between 1 and pq where p and q are two
distinct primes. We choose one of them, all with equal
probability.
a) What is the probability that we choose any given number?
b) What is the probability that we choose a number that is
i) relatively prime to p?
ii) relatively prime to q?
iii) relatively prime to pq?

Write public and private key where p = 13, q = 37 and
n = 481. Totient is 432. E is 19 and d is 91. Write public key as
(n = , e =) and private key as (n = , d =)

Sociology Multiply choice
1) The concept of a (n) _______________________
group refers to those who share a collective identity based on
common ancestry and cultural characteristics such as language,
foodways, and religion; whereas, _________________________ is
concept that refers to a socially constructed category that is
generally based on physical and other biological characteristics
defined as important in the society.
A)
mulatto; ethnic
B)
ethnic; race
C)
racial; ethnic
D)
white ethnics; mulatto
2)Using Karl Marx’s language, the relatively
small group...

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