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.
FALSE: For all integers and ,
Statement
False claim: for all integers and ,
(Common divisor, and the greatest common divisor , with the convention , Common multiple, and the least common multiple , taken to be when or ).
The true statement is Every common multiple of and is a multiple of , and , with an absolute value on the right: . The two differ as soon as is negative, and is a witness: there and , so the left side is , while .
Numerals. For the symbol inside means , the embedding of The naturals embed in the integers.
Facts & Assumptions
Given: The integers , and .
is a commutative ring: multiplication is associative and commutative, , , multiplication distributes over addition, and every has an additive inverse (The integers form a commutative ring, Arithmetic on the integers, The integers as equivalence classes of pairs of naturals).
The order on is total, antisymmetric and transitive and is compatible with addition (The integers form a totally ordered ring, Order on the integers).
exactly when , , , and every common divisor of and divides (Every common divisor of and divides ; consequently exactly when , , , and every common divisor of and divides — a characterisation that holds at as well).
If and then for all ; and for every (Divisibility is reflexive and transitive on , and is linear: if and then for all integers ; also implies , and , Divisibility in : when for some integer ).
is injective with image the nonnegative integers, and preserves addition and multiplication (The naturals embed in the integers).
The refuted claim: for all integers .
Refutation
. Indeed , and and by [L5]; and , so every common divisor of and divides by [L5]. By [L4] this characterises .
, and since and .
: otherwise , that is , whence in , contradicting injectivity of .
By [L3], ; with step 1.1 this reads , so by cancellation of the nonzero factor .
Therefore , while , and these differ: [L9] is false at .
Remarks
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Every pair with refutes it, not just this one: and are both nonnegative by construction (Common divisor, and the greatest common divisor , with the convention , Common multiple, and the least common multiple , taken to be when or ), so their product is nonnegative, while is negative. The witness above is simply a small instance.
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The claim is true when and are both nonnegative, which is why it is a natural slip: in that case and the two statements coincide. Every common multiple of and is a multiple of , and is the version that holds for all integers.
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The absolute value is not the only convention doing work. At the true identity reads , which holds only because was fixed in Common multiple, and the least common multiple , taken to be when or .
Depends on
- Every common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a,b)$, and $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = |ab|$
- Common multiple, and the least common multiple $\operatorname{lcm}(a,b)$, taken to be $0$ when $a = 0$ or $b = 0$
- Common divisor, and the greatest common divisor $\gcd(a,b)$, with the convention $\gcd(0,0) := 0$
- Every common divisor of $a$ and $b$ divides $\gcd(a,b)$; consequently $d = \gcd(a,b)$ exactly when $d \ge 0$, $d \mid a$, $d \mid b$, and every common divisor of $a$ and $b$ divides $d$ — a characterisation that holds at $(a,b) = (0,0)$ as well
- Divisibility is reflexive and transitive on $\mathbb{Z}$, and is linear: if $d \mid a$ and $d \mid b$ then $d \mid ax + by$ for all integers $x, y$; also $d \mid a$ implies $d \mid ac$, $-d \mid a$ and $d \mid -a$
- Divisibility in $\mathbb{Z}$: $d \mid a$ when $a = dq$ for some integer $q$
- The absolute value $|a|$ of an integer
- Absolute value in $\mathbb{Z}$: $|a| \ge 0$; $|a| = 0$ exactly when $a = 0$; $|-a| = |a|$; $|ab| = |a|\,|b|$; $-|a| \le a \le |a|$; and $|a| \le c$ exactly when $-c \le a \le c$
- The integers have no zero divisors; multiplicative cancellation
- The integers form a commutative ring
- The integers form a totally ordered ring
- Arithmetic on the integers
- Order on the integers
- The naturals embed in the integers
- The integers as equivalence classes of pairs of naturals
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
40 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
- Least common multiple (Wikipedia) (standard reference, not scraped)
- Greatest common divisor (Wikipedia) (standard reference, not scraped)