NP-completeness conditions for some types of systems of linear Diophantine dis-comparisons consistency checking

Two series of number-theoretic problems concerning modulo m dis-comparisons with explicitly pointed out parameters are proposed in the paper. Conditions upon the parameters implying that every problem of a series is an NP-complete one are proved. It is proved that for every m > 2 the consistency problem for a system of modulo m dis-comparisons every of which contains exactly 3 variables (even if every coefficient belongs to {−1, 1}) is NP-complete. It is also proved that for every m > 3 the consistency problem for a system of modulo m dis-comparisons every of which contains exactly 2 variables is NP-complete. If P =NP then the statement of the proved theorem can not be changed by means of replacing the term «2-dis-comparison» by the term «1-dis-comparison». Refs 6