FLINT3.7.0-dev
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_padic_exp_bound

Synopsis#include <flint/padic.h>
slong _padic_exp_bound(slong v, slong N, const fmpz_t p)source

Returns nn such that ord⁡p(xj/j!)≥N\operatorname{ord}_p(x^j / j!) \ge N for all j≥nj \ge n and all xx with ord⁡p(x)=v\operatorname{ord}_p(x) = v, so that the exponential series can be truncated after nn terms. Requires 1≤v<N1 \le v < N for odd pp and 2≤v<N2 \le v < N for p=2p = 2.

If pp fits in a slong the result is ⌈((p−1)N−1)/((p−1)v−1)⌉\lceil ((p - 1)N - 1) / ((p - 1)v - 1) \rceil, which follows from ord⁡p(j!)≤(j−1)/(p−1)\operatorname{ord}_p(j!) \le (j - 1)/(p - 1); otherwise it is ⌈N/v⌉\lceil N / v \rceil, since then j!j! has no factor pp for the jj that occur.

Cite this page

The FLINT team, “_padic_exp_bound (padic.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/padic/_padic_exp_bound.html (generated 2026-11-02).