Worksheet 1
Remember, no credit will be given for answers without mathematical or logical justification.
Part I
1) Note that 439 068 = 12 Ć36589. If it is 7 oāclock right now, what time will it be in
439 069 hours?
For problems 2 and 3, we make the following definition. A number pis prime
in mod narithmetic if the only way that
pā”aĆb(mod n) (1)
is that either aā”1 (mod n)and bā”p(mod n), or aā”p(mod n)and bā”1 (mod n).
2) In arithmetic mod 7, are there any primes? For instance, 3 ā”2Ć5 (mod 7) so 3 is not
prime.
3) Can you prove that, in mod 6 arithmetic, 5 is prime? Can you show that both 1 and
2 are not prime?
Part II
4) Consider the M-sequence (Mis for Math 170), given by M1= 1, M2= 1 and for any
n > 2
Mn= 2Mnā1+ 3Mnā2.(2)
List the first 8 M-numbers.
5) Consider the nth ratio of M-numbers, defined to be
Rn=Mn+1
Mn
.
Determine a recursive formula for the Rn.
6) Obviously the Mnumbers are getting big fast. But how fast? Determine, in the limit
as ngets bigger, what the ratio Rnof successive M-numbers is.
Part III
7) A dyadic rational is a fraction whose denominator is a power of two. In other words,
a fraction of the form
m
2n(3)
where mand nare integers. Which of the following numbers are dyadic rationals?
Ļ, 1
2,ā2,3
8,8
3,17
16,9,1
48 (4)
8) Construct a one-to-one correspondence between the positive dyadic rationals and the
natural numbers. Make a list of the first 13 dyadic numbers, according to your corre-
spondence. (As in class, you may use a picture to describe your method, as long as it
is clear.)
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