Download Module 5 homework

Module 5 Homework
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(651 PGH – 8am to 5pm)
79 point assignment
1
5 points
1. Find two examples where for 4 distinct numbers a, b, c, and d:
a 2  b2
2
c2  d 2
What tips can you provide for others on how to find the numbers?
2
5 points
2.
Show with examples and discuss why it’s true:
A.
The product of any 3 consecutive natural numbers is divisible by 6.
B.
The product of any 4 consecutive natural numbers is divisible by 24.
C.
The product of any 5 consecutive natural numbers is divisible by 120.
You must use 12 different numbers in the example. If you use 5, you may only use it in
one example.
Fact:
The product of n consecutive integers is divisible by n!
D.
Why is this fact true?
3
10 points
3.
Create numbers of the form:
22  3
n
Calculate the numbers for n = 1, 2, 3, 4:
For which n does the number exceed the limit on number gossip?
Which of the following vocabulary words (from “number gossip”) apply to each
number?
even
odd
algebraic
transcendental
square-free
deficient,
abundant,
perfect,
prime
happy
unhappy
4
5 points
4.
Analyze the following patter
1x1
11 x 11
111 x 111
1111 x 1111
(R 5 ) 2
=
=
=
=
=
1
121
12321
1234321
123454321
(R 9 ) 2
=
___________________________
The answers are consecutive integer palindromes. What does that sentence say in
regular English?
Write 2 brief paragraphs about why this works from R1 to R9.
5
5 points
5.
Find a pair of twin primes other than the one in the example.
Note that they are of the form (n  1, n + 1) with the composite number, n,
between them.
(example, 5 and 7, n = 6).
Now show that
2n 2  2 can be represented as the sum of 2, 3, 4, and 5 perfect squares (not
necessarily distinct summands here)
example: representing 2(36) +2 = 74 as two perfect squares: 25 + 49.
Pick a prime number > 25 and call it p?
Does the same fact, this time with 2p 2  2 work like this?
Show that it might or might not – trade this around with study buddies to find
some of each.
6
6 points
6.
Theorem:
If n divides m, the Rn divides Rm.
Where Rn is the repunit with n 1’s.
Use this theorem to find factors of R10. Then factor R10 to primes.
7
6 points
7.
Find a set of prime numbers whose difference is 3. Are there many of these or
few?
8
6 points
8.
A prime triplet is {p, p + 2, p + 6}. when p and p + 2 are twin primes. Give 3
examples of 3 prime triplets.
Why does this formula work?
hint: see primes and mod 6 from Module 2
9
6 points
9.
Using the prediction formula for the number of primes fewer than a given x,
x
N=
, find the predicted number of primes below 50. Now, using a list of
ln x
primes to count (easily available on the internet), how many are actually between
1 and 50?
10
8 points
10.
Theorem:
If p and p2 + 8 are prime, then so is p3 + 4.
Find two examples of this theorem or one example and an explanation of why
there’s not a second example.
11
3 points
11.
Find a gap of length at least 7 between two primes using the way I showed you in
class.
What are the two primes?
12
4 points
12.
Give an arithmetic sequence with a leading term a and a difference d that are
relatively prime. Show that it has many primes.
Give an arithmetic sequence with a first term a and a difference d that are not
relatively prime. Show that it’s prime free.
13
10 points
13.
A.
B.
List 5 ways in which lucky numbers and primes are alike.
List 5 ways in which lucky numbers and primes are different.
Use good grammar and spelling. Make sure a reasonable person can understand what
you’re saying.
14