The authors consider the difference equations
with delay matrix
where for . The authors put in the case and note that the characteristic polynomial for the equation (1) is
For the polynomial the authors obtain the following result:
Theorem 4. (i) Suppose The characteristic polynomial has a zero which is a primitive -th root of unity if and only if
(ii) Any primitive root of unity which satisfies must be a primitive -th root of unity for some In addition each root of of modulus one has multiplicity 1.
In the case , i.e for the equation
the authors obtain the following results.
Theorem 7. Let and . Then all integer solutions to equation (3) must have period .
Theorem 8. Let and . Then there exists a nontrivial integer solutions to equation (3) which has period . The authors give an effective construction of such a solution.
Theorem 9. Let and Then there exists a periodic integer solution to equation (3) with minimal period .
Theorem 11. Let and Then any integer solutions to the equation (3) must satisfy the inequality
Theorem 12. Let and . Then there exists an integer solution to the equation (3) such that
As a corollary of Theorems 7, 9, and 12 the authors prove the following
Theorem 10. Let . . Then all integer solutions to equation (3) have period if and only if . All constructions are effective.
Unexpected results, detailed proofs, supplementary material (interesting observations, a Conjecture, some Open Questions) – all this left a very good impression (some stylistic bugs and misprints, which do not deserve to be mentioned, cannot destroy it).