MathDB

Let

f(x)

be a periodic function with periods

T

and

1

(

0<T<1

).Prove that: (1)If

T

is rational,then there exists a prime

p

such that

\frac{1}{p}

is also a period of

f

; (2)If

T

is irrational,then there exists a strictly decreasing infinite sequence {

a_n

},with

1>a_n>0

for all positive integer

n

,such that all

a_n

are periods of

f

2