Pythagorean fuzzy prioritized aggregation operators with priority degrees for multi-criteria decision-making

Purpose-The authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees.The properties of the existing method are routinely compared to those of other current approaches,emphasizing the superiority of the presented work over currently used methods.Furthermore,the impact of priority degrees on the aggregate outcome is thoroughly examined.Further,based on these operators,a decision-making approach is presented under the Pythagorean fuzzy set environment.An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.Design/methodology/approach-In real-world situations,Pythagorean fuzzy numbers are exceptionally useful for representing ambiguous data.The authors look at multi-criteria decision-making issues in which the parameters have a prioritization relationship.The idea of a priority degree is introduced.The aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels.Consequently,the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees.Findings-The authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees.The properties of the existing method are routinely compared to those of other current approaches,emphasizing the superiority of the presented work over currently used methods.Furthermore,the impact of priority degrees on the aggregate outcome is thoroughly examined.Further,based on these operators,a decision-making approach is presented under the Pythagorean fuzzy set environment.An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.Originality/value-The aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels.Consequently,the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees.The properties of the existing method are routinely compared to those of other current approaches,emphasizing the superiority of the presented work over currently used methods.Furthermore,the impact of priority degrees on the aggregate outcome is thoroughly examined.

A Remark on Jesmanowicz' Conjecture for the Non-coprimality Case

Let a, b, c be relatively prime positive integers such that a^2＋ b^2= c^2. Jesmanowicz＇conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation（aN）x＋（b N）y=（cN）zhas no positive solution（x, y, z） other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.