摘要
The hardest step to solve Hilbert's tenth problem is to prove that the exponential rela-tion is Diophantine. In the study of decision problems concerning the solvability of Diophan-tine equations with few unknowns, reducing unknowns in Diophantine representations playsan important role. In this paper, we give Diophantine representations of C=φ_B(A,1) (whereφ_0(A,1) =0, φ_1(A,1) =1, φ_(m+1)(A,1) =Aφ_m(A,1)--φ_(m-1)(A,1)) and W =V~∧A_1,…,A_k∈□∧S|T∧R>0 with only 3 and 5 natural number unknowns respectively, C =φ_B(A,1) (on thecondition 1<|B|<|A|/2-1) and W =V^B∧A_1,…,A_k∈□∧S|T with 4 and 6 integer unknownsrespectively.
The hardest step to solve Hilbert’s tenth problem is to prove that the exponential rela-tion is Diophantine. In the study of decision problems concerning the solvability of Diophan-tine equations with few unknowns, reducing unknowns in Diophantine representations playsan important role. In this paper, we give Diophantine representations of C=φ<sub>B</sub>（A,1） (whereφ<sub>0</sub>（A,1） =0, φ<sub>1</sub>（A,1） =1, φ<sub>m+1</sub>（A,1） =Aφ<sub>m</sub>（A,1）--φ<sub>m-1</sub>（A,1）) and W =V<sup>∧</sup>A<sub>1</sub>,…,A<sub>k</sub>∈□∧S|T∧R>0 with only 3 and 5 natural number unknowns respectively, C =φ<sub>B</sub>（A,1） （on thecondition 1<|B|<|A|/2-1） and W =V<sup>B</sup>∧A<sub>1</sub>,…,A<sub>k</sub>∈□∧S|T with 4 and 6 integer unknownsrespectively.
基金
Project supported by the National Natural Science Foundation of China.
