Download I. Precisely complete the following definitions: 1. A natural number n

I. Precisely complete the following definitions:
1. A natural number n is composite whenever ...
See class notes for the precise definitions
2. Fix n in N. The number s(n) represents ...
3. For A and B sets, we say f is a function from A to B when ...
II. Give an example of a function from N to N that is onto and assigns every natural
number to a natural number different from itself. Explain why your function is onto in
five [5] sentences or less.
Try the assignment f(1) = 2; f(2) = 1; f(3) = 4; f(4) = 3; … .
More precisely, but not necessary for credit, f(m) = m + 1 if m is odd and f(m) = m –
1 if m is even.
This is a function; and given any natural n, we can find something that f assigns to n.
If n is even f(n-1) = n; if n is odd f(n+1) = n.
III. Consider the diophantine equation 1694X + 4060Y = 56.
1. In five [5] sentences or less, outline the method developed in this course to
determine a solution to this equation.
Use the Euclidean Algorithm to find d = (1694,4060) and check that this divides
56. Then use this algorithm to express d as a linear combination of 1694 and
4060. Take this express and multiply through by an appropriate integer (the
integer that when multiplied by d gives 56) to find a solution.
2. Find all the solutions to this equation.
Check d = (1694,4060) = 14. The linear combination that emerges from the
Euclidean Algorithm is (1694)(-139) + (4060)(58) = 14. Multiplying through by 4
gives X = (4)(-139) = -556 and Y = (4)(58) = 232.
Now by Thm 2.2 all solutions are given by
X = (-556) + (290)t and Y = 232 – (121)t
where t is an integer.
IV. Which number has more positive divisors: 1848 or 1850? Justify your choice in five
[5] sentences or less.
Note 1848 = (23)(3)(7)(11) and 1850 = (2)(52)(37). Computing nu values, 1848 has
more divisors.
NB. The number of distinct primes in the factorization of a number alone does not
tell you how many divisors it can have. For example: 1848 has more distinct prime
divisors than 240, but 240 has more divisors!
V. Prove the following for all natural numbers n: 1 + 3 + 5 + … + (2n – 1) = n2.
Standard proof using PMI.1. Be sure that your argument doesn’t contain any
statement that suggests the result you are attempting to prove!
VI. Let a, b, c be in Z with (a,b) = 1. Assume c = aq and c = br for some integers q and
r.
1. Prove that there is an integer m for which r = am.
Go with a direct proof: beginning with the observation that you can express 1 as
a linear combination of a and b. Multiply this expression by r and manipulate
the new expression using appropriate substitutions to see r as a multiple of a.
There’s your m!
2. Prove this m also gives q = bm.
Take the expression for m found above and check bm gives q.
VII. Only one of the following will be graded. Please circle the number of the statement
you want me to consider.
1. Let p be prime and a, b, m, n be naturals. We say pm exactly divides a when pm
divides a, but pm+1 does not. Let pm exactly divide a and pn exactly divide b where m ≠
n. What power of p exactly divides a + b. Prove your claim.
2. Prove there are infinitely many primes of the form 6k + 5 with k > 0.
1. The power you want is the minimum of the numbers m and n. Prove this
power of p divides a + b by a direct argument. Show no larger power of p
divides a + b by contradiction.
2. See the proof by contradiction for the theorem proved in class: there are
infinitely many primes of the form 4k + 3. There are only a couple modifications
of this argument required to show that there are infinitely many primes of the
form 6k + 5.