MATH 135, Fall 2010
Solution of Assignment #9
Problem 1
.
Suppose that
p
and
q
are prime numbers with
p > q
. Suppose also that
n
=
pq
and
φ
(
n
) = (
p

1)(
q

1).
(a) Prove that
p
+
q
=
n

φ
(
n
) + 1 and
p

q
=
p
(
p
+
q
)
2

4
n
.
(b) If
n
= 1 281 783 203 and
φ
(
n
) = 1 281 711 600, determine
p
and
q
.
Solution.
(a) In the ﬁrst equation,
RS =
n

φ
(
n
) + 1 =
pq

(
p

1)(
q

1) + 1 =
pq

(
pq

p

q
+ 1) + 1 =
p
+
q
= LS
In the second equation,
RS =
p
(
p
+
q
)
2

4
n
=
p
p
2
+ 2
pq
+
q
2

4
pq
=
p
p
2

2
pq
+
q
2
=
p
(
p

q
)
2
=
p

q
= LS
since
p > q
.
(b) From (a),
p
+
q
=
n

φ
(
n
) + 1 = 1 281 783 203

1 281 711 600 + 1 = 65 604
and
p

q
p
(
p
+
q
)
2

4
n
=
p
65 604
2

4(1 281 783 203) =
√
4 = 2
Thus, 2
p
= (
p
+
q
) + (
p

q
) = 65 606 whence
p
= 32 803.
Lastly,
q
= (
p
+
q
)

p
= 98 096

74 623 = 32 801.
Problem 2
.
Use Fermat’s Little Theorem and the Square and Multiply Algorithm to show that the integer
3599 is not prime, without testing each prime
p
≤
√
3599 to see if is a factor. (Hint: compute 2
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 Spring '08
 ANDREWCHILDS
 Prime Numbers, Prime number, Power of two

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