The Real And Complex Number Systems

Integers

1.1 Prove that there is no largest prime.
Proof : Suppose p is the largest prime. Then p! + 1 is NOT a prime. So, there exists a prime q such thatq |p! + 1 ⇒ q |1 which is impossible. So, there is no largest prime. Remark: There are many and many proofs about it. The proof that we give comes from Archimedes 287-212 B. C. In addition, EulerLeonhard (1707-1783) find another method to show it. The method is important since it develops to study the theory of numbers by analytic method. The reader can see the book, An Introduction To The TheoryOf Numbers by Loo-Keng Hua, pp 91-93. (Chinese Version)

1.2 If n is a positive integer, prove the algebraic identity
n−1

a − b = (a − b)
k=0

n

n

ak bn−1−k

Proof : It suffices to showthat
n−1

x − 1 = (x − 1)
k=0

n

xk .

1

Consider the right hand side, we have
n−1 n−1 n−1

(x − 1)
k=0

x =
k=0 n

k

x

k+1



xk

=
k=1 n

xk −

k=0 n−1 kx

k=0

= x − 1.

− 1 is a prime, prove that n is prime. A prime of the form 2 − 1, where p is prime, is called a Mersenne prime.
p

1.3 If 2

n

Proof : If n is not a prime, then say n= ab, where a > 1 and b > 1. So, we have
b−1

2ab − 1 = (2a − 1)
k=0

(2a )k

which is not a prime by Exercise 1.2. So, n must be a prime. Remark: The study of Mersenne prime is important; itis related with so called Perfect number. In addition, there are some OPEN problem about it. For example, is there infinitely many Mersenne nembers? The reader can see the book, An Introduction To TheTheory Of Numbers by Loo-Keng Hua, pp 13-15. (Chinese Version)

1.4 If 2
form 2 So,
2m

+ 1 is a prime, prove that n is a power of 2. A prime of the + 1 is called a Fermat prime. Hint. Useexercise 1.2.

n

Proof : If n is a not a power of 2, say n = ab, where b is an odd integer. 2a + 1 2ab + 1 and 2a + 1 < 2ab + 1. It implies that 2n + 1 is not a prime. So, n must be a power of 2.... [continua]

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