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S^{n+k}(a,b,c) = S^{n}(a,b,c) (mod abc)

Source: Turkey TST 2001 - P3

April 5, 2013

modular arithmeticparameterizationnumber theory proposednumber theory

Problem Statement

For all integers

x,y,z

, let

S(x,y,z) = (xy - xz, yz-yx, zx - zy).

Prove that for all integers

a

,

b

and

c

with

abc>1

, and for every integer

n\geq n_0

, there exists integers

n_0

and

k

with

0<k\leq abc

such that

S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.

(

S^1 = S

and for every integer

m\geq 1

,

S^{m+1} = S \circ S^m.

(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).

)

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