Question

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.

Answer #1

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,...

Let
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that
(a) There exist two elements of S that are relatively prime,
and
(b) There exist two elements of S, one of which divides the
other.

1. Let n be an odd positive integer. Consider a list of n
consecutive integers.
Show that the average is the middle number (that is the number
in the
middle of the list when they are arranged in an increasing
order). What
is the average when n is an even positive integer instead?
2.
Let x1,x2,...,xn be a list of numbers, and let ¯ x be the
average of the list.Which of the following
statements must be true? There might...

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...

A positive integer n is called "powerful" if, for every
prime factor p of n, p2 is also a
factor of n. An example of a powerful number is
A) 240
B) 297
C) 300
D) 336
E) 392

In number theory, Wilson’s theorem states that a natural number
n > 1 is prime
if and only if (n − 1)! ≡ −1 (mod n).
(a) Check that 5 is a prime number using Wilson’s theorem.
(b) Let n and m be natural numbers such that m divides n. Prove the
following statement
“For any integer a, if a ≡ −1 (mod n), then a ≡ −1 (mod m).”
You may need this fact in doing (c).
(c) The...

Let λ be a positive irrational real number. If n is a positive
integer, choose by the Archimedean Property an integer k such that
kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all
φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the
proof of the density of the rationals in the reals.)

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....

1. Let p be any prime number. Let r be any integer such that 0
< r < p−1. Show that there exists a number q such that rq =
1(mod p)
2. Let p1 and p2 be two distinct prime numbers. Let r1 and r2 be
such that 0 < r1 < p1 and 0 < r2 < p2. Show that there
exists a number x such that x = r1(mod p1)andx = r2(mod p2).
8. Suppose we roll...

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