A product expansion in 𝑝-adic and other non-Archimedean fields

Knopfmacher, Arnold; Knopfmacher, John

doi:10.1090/S0002-9939-1988-0929409-9

A product expansion in $p$-adic and other non-Archimedean fields
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by Arnold Knopfmacher and John Knopfmacher PDF

Proc. Amer. Math. Soc. 104 (1988), 1031-1035 Request permission

Abstract:

An algorithm is introduced and shown to lead to a unique infinite product representation for a given $p$-adic integer $A$ with leading coefficient 1 as a product \[ A = \prod \limits _{n = 1}^\infty {(1 + {b_n}{p^{{r_n}}})} \] where $1 \leq {b_n} \leq p - 1,{r_n} \in {\mathbf {N}}$ and ${r_{n + 1}} > {r_n}$. The degree of approximation by the natural number $(1 + {b_1}{p^{{r_1}}}) \cdots (1 + {b_n}{p^{{r_n}}})$ is also considered. In addition we derive similar representations for elements of arbitrary complete non-Archimedean fields with discrete valuations.

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