Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques

Table 1

Comparison of HL, AM/VN, and HM/BM.

Methods	𝑥𝑘  (mod𝑝^𝛼)	𝑥𝑘+1  (mod𝑝^𝑚𝛼)
HL	3	3 (mod7^2)
AM/VN	3	199 (mod7^3)
HM/BM	3	1914 (mod7^4)
HL	3	199 (mod7^3)
AM/VN	199	37399890 (mod7^9)
HM/BM	1914	24211798063820 (mod7^16)
HL	199	1914 (mod7^4)
AM/VN	37399890	43741341794232381830191 (mod7^27)
HM/BM	24211798063820	954381941076…900159613690770 (mod7^64)
HL	1914	573 (mod7^5)
AM/VN	43741341794232381830191	24452773833…232537600 (mod7^81)

Solution of the congruence 2𝑥³≡5(mod7⁸¹) with initial estimate 𝑥₁=3.