Question

A positive integer *n* is called "powerful" if, for every
prime factor *p* of *n*, p^{2} is also a
factor of *n.* An example of a powerful number is

A) 240

B) 297

C) 300

D) 336

E) 392

Answer #1

Show that if p is a positive integer such that both p and
p2 + 2 are prime, then p = 3.

Let a positive integer n be called a super exponential number if
its prime factorization contains at least one prime to a power of
1000 or larger. Prove or disprove the following statement: There
exist two consecutive super exponential numbers.

Activity 6.6.
(a)
A positive integer that is greater than 11 and not
prime is called composite.
Write a technical definition for the concept of composite number
with a similar level of detail as in the “more complete” definition
of prime number.
Note.
A number is called prime if its only divisors are 1 and
itself.
This definition has some hidden parts: a more complete
definition would be as follows.
A number is called prime if
it is an integer,...

Prove every integer n ≥ 2 has a prime factor. (You cannot just
cite the Funda- mental Theorem of Arithmetic; this was the first
step in proving the Fundamental Theorem of Arithmetic

Let n be a positive integer. Show that every abelian group of
order n is cyclic if and only if n is not divisible by the square
of any prime.

Assume that p does not divide n for every prime number
p with n> 1 and p <= (n) ^ (1/3).
Then prove that n is a prime number or a product of two prime
numbers

A positive integer is called a novenary if all of its prime
factors are less than or equal to 9. Find two sets A and B of
distinct novenary numbers so that if you sum the square roots of
the numbers in A and subtract the sum of the square roots of the
number in B the answer is close to zero.

4. Prove that if p is a prime number greater than 3, then p is
of the form 3k + 1 or 3k + 2.
5. Prove that if p is a prime number, then n √p is irrational
for every integer n ≥ 2.
6. Prove or disprove that 3 is the only prime number of the form
n2 −1.
7. Prove that if a is a positive integer of the form 3n+2, then
at least one prime divisor...

/*
This program should check if the given integer number is
prime.
Reminder, an integer number greater than 1 is prime if
it divisible only by itself and by 1.
In other words a prime number divided by any other natural
number
(besides 1 and itself) will have a non-zero remainder.
Your task:
Write a method called checkPrime(n) that will
take
an integer greater than 1 as an input, and return true
if that integer is prime; otherwise, it should...

An
integer 'n' greater than 1 is prime if its only positive divisor is
1 or itself. For example, 2, 3, 5, and 7 are prime numbers, but 4,
6, 8, and 9 are not. Write a python program that defines a function
isPrime (number) with the following header: def isPrime (number):
that checks whether a number is prime or not. Use that function in
your main program to count the number of prime numbers that are
less than 5000....

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