NP completeness Conditions for Verifying the Consistency of Several Kinds of Systems of Linear Diophantine Discongruences

摘要

We propose two series of number-theory problems with explicitly marked out parameters related to discongruences modulo m. We find parameter constraints that provide the NP completeness for any problem of every series. For any m > 2, we prove the NP completeness of the verification problem for the consistency of a system of linear discongruences modulo m such that any discongruence contains exactly three variables, including the case where its coefficients belong to {-1, 1}. For any m > 3, we prove the NP completeness of the verification problem for the consistency of a system of linear discongruences modulo m such that any discongruence contains exactly 2 variables. If P ≠ NP, then one cannot change the term 2-discongruence for the term 1-discongruence in the statements of the proven theorems.