(4 marks)
(iii)
Let
Φ(
X
)
be the polynomial
X
4
+
X
3
+
X
2
+
X
+ 1
.
(a)
Show that the set of primes
p
for which there is an integer
n
with
p

Φ(
n
)
is infinite.
(3 marks)
(b)
Now let
a
be an integer, and let
p >
5
be a prime dividing
Φ(
a
)
.
Show that
p
divides
a
5

1
,
and deduce that the residue class of
a
in
(
Z
/p
Z
)
×
,
the unit
group of
Z
/p
Z
,
has order
5
.
(5 marks)
(c)
Now deduce that
5

p

1
,
and conclude that there are infinitely
many primes of the form
5
n
+ 1
.
(3 marks)
3
(i)
Show that
b
x
+
y
c  b
x
c  b
y
c
= 0
or
1
for all
x, y
∈
R
.
(Recall that
b
x
c
is the greatest integer not exceeding
x.
)
(3 marks)
Now let
n
be a positive integer, and let
p
be a prime.
(a)
Write down a formula for the highest power of
p
that divides
n
!
,
and calculate the number of zeros at the end of
2009!
.
(4 marks)
(b)
Show that if
p
r
divides
2
n
n
¶
then
p
r
≤
2
n.
(4 marks)
(c)
Show that if
n
≥
3
and
2
n/
3
< p
≤
n,
then
2
n
n
¶
is not divisible
by
p.
(4 marks)
(ii)
State Bertrand’s Postulate.
(1 mark)
Prove the following statements.
(a)
If
n
is a positive integer then the binomial coefficient
2
n
n
¶
is not
a square.
(3 marks)
(b)
Call a positive integer
n
a
primeone
if either
n
= 1
or
n
is a prime
number. Then every positive integer can be written as a sum of distinct primeones.
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 Fall '13
 430
 Statistics, Number Theory, lim, Prime number, Divisor, Riemann zeta function

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