Question

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 of a is of the form 3n + 2.

Answer #1

Prove that every prime greater than 3 can be written in the form
6n+ 1 or 6n+ 5 for some positive integer n.

Prove that every prime greater than 3 can be written in the form
6n + 1 or 6n + 5 for some positive integer n.

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,...

8. Prove or disprove the following statements about
primes:
(a) (3 Pts.) The sum of two primes is a prime number.
(b) (3 Pts.) If p and q are prime numbers both greater than 2,
then pq + 17 is a composite number.
(c) (3 Pts.) For every n, the number n2 ? n + 17 is always
prime.

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....

A prime number is an integer greater than 1 that is evenly
divisible by only 1 and itself. For example, 2, 3, 5, and 7 are
prime numbers, but 4, 6, 8, and 9 are not. Create a PrimeNumber
application that prompts the user for a number and then displays a
message indicating whether the number is prime or not. Hint: The %
operator can be used to determine if one number is evenly divisible
by another. ( Java programing...

A prime number (or a prime) is an integer greater than
1 that is not a product of two smaller integer. Created a program
on visual studio named PrimeNumberTest that does
the following:
1) prompt the user for input of an integer
2) test if the integer is a prime number
3) display the test result

Prove that a natural number m greater than 1 is prime if m has
the property that it divides at least one of a and b whenever it
divides ab.

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.

Use Mathematical Induction to prove that 3n < n! if n is an
integer greater than 6.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 12 minutes ago

asked 20 minutes ago

asked 48 minutes ago

asked 51 minutes ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago