2

Part of 1997 Turkey Team Selection Test

Problems(2)

Turkey TST 1997 Problem 2, how many pairs

Source: Turkey TST 1997 Problem 2

(a_{n})

(b_{n})

are deﬁned by

a_{1} = \alpha

b_{1} = \beta

a_{n+1} = \alpha a_{n} - \beta b_{n}

b_{n+1} = \beta a_{n} + \alpha b_{n}

n > 0.

(\alpha, \beta)

of real numbers are there such that

a_{1997} = b_{1}

b_{1997} = a_{1}

algebra proposedalgebra

Turkey TST 1997 Problem 5, there exist n, x_i and y_i

Source: Turkey TST 1997 Problem 5

Show that for each prime

p \geq 7

, there exist a positive integer

n

x_{i}

y_{i}

(i = 1, . . . , n)

, not divisible by

p

x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}

x_{n+1} = x_{1}

modular arithmeticabstract algebranumber theory proposednumber theory