Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
If is prime and then
Statement
Let be prime. If , then
Facts & Assumptions
Given: a prime number and integers with .
The closed formula gives ( for ; hence , the quotient is a natural number, and )
For a nonzero integer , exactly when (For a prime and a nonzero integer : and ; holds exactly for ; exactly when ; ; and ).
The -adic valuation is additive on products of nonzero integers ( for nonzero integers , and whenever , and are all nonzero).
Proof
The identity of [F1] gives
Every factor in each of , , and lies strictly between and , so [F2] gives valuation for each factor. Repeated additivity from [F3] therefore gives
Substituting step 1.2 into step 1.1 gives , which is exactly the statement that .
Depends on
- $\binom{n}{k}\,k!\,(n-k)! = n!$ for $k \le n$; hence $\binom{n}{k}\,k! = n^{\underline{k}}$, the quotient $n!/(k!(n-k)!)$ is a natural number, and $\binom{n}{k} = \binom{n}{n-k}$
- The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$
- The $p$-adic valuation $v_p(a)$ of a nonzero integer: the greatest $k \in \mathbb{N}$ with $p^{k} \mid a$
- For a prime $p$ and a nonzero integer $a$: $p^{v_p(a)} \mid a$ and $p^{v_p(a)+1} \nmid a$; $p^{k} \mid a$ holds exactly for $k \le v_p(a)$; $v_p(a) \ge 1$ exactly when $p \mid a$; $v_p(1) = v_p(-1) = 0$; and $v_p(p) = 1$
- $v_p(ab) = v_p(a) + v_p(b)$ for nonzero integers $a, b$, and $v_p(a+b) \ge \min\{v_p(a), v_p(b)\}$ whenever $a$, $b$ and $a+b$ are all nonzero
- Euclid's lemma: if $p$ is prime and $p \mid ab$ then $p \mid a$ or $p \mid b$
- Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$
- The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
Used by
Dependency tree · two levels
61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- N. Alon, Combinatorial Nullstellensatz, proof of Theorem 3.2 (standard reference, not scraped)