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  "body": "\n## Summary\n\nThe $p$-adic gamma function $\\Gamma_p$ of Morita is a continuous function on the [[p-adic integers|V6TJAFZ4MD]] that plays the role of the [[gamma function|0H5SYBSED9]].\nIt interpolates a modified [[factorial|0XWG0JGDFW]] in which the factors divisible by $p$ are left out.\nIt satisfies a functional equation like $\\Gamma(x+1) = x\\,\\Gamma(x)$, and it appears in the Gross–Koblitz formula for Gauss sums.\n\n## Explanations\n\n### L1 human\n\nThe ordinary factorial $n!$ cannot be extended continuously in the $p$-adic sense, because $n!$ becomes divisible by high powers of $p$ and so $p$-adically small.\nThe fix is to skip the factors that are multiples of $p$.\nThe resulting products, with a sign, change only a little when $n$ changes by a high power of $p$.\nSo they extend to a continuous function on all $p$-adic integers.\nFor example, for $p = 3$ the value at $5$ is $-(1 \\cdot 2 \\cdot 4) = -8$, since the factor $3$ is skipped.\n",
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