Do coprime amicable pairs exist? Search for $(m,n)$ with $\gcd(m,n)=1$ and $\sigma(m)=\sigma(n)=m+n$ (Guy UPINT §B4)
Statement
An amicable pair $(m,n)$ with $m\ne n$ satisfies $s(m)=n$ and $s(n)=m$, equivalently $\sigma(m)=\sigma(n)=m+n$. Does there exist an amicable pair whose two members are coprime, $\gcd(m,n)=1$? Exhibit such a pair, or verify that none exists below a stated bound.
Acceptance. FULLY RESOLVES: a coprime amicable pair — two integers $m,n$ each with a prime factorization (so $\sigma$ is machine-checkable), with $\gcd(m,n)=1$ verified and $\sigma(m)=\sigma(n)=m+n$. ADVANCES: certify that no coprime amicable pair has $\min(m,n)\le B$ for a new bound $B$ (reproducible exhaustive search with a verifier), or establish a new structural lower bound (e.g. raise the $10^{67}$ product bound or the distinct-prime count) with a checkable computation. Deliver the certificate and verifier.
Background
Guy, 'Unsolved Problems in Number Theory' (3rd ed.), §B4 (amicable numbers). No coprime amicable pair is known. Any such pair would have to satisfy strong constraints: its product must exceed $10^{67}$ and it must have at least seven distinct prime factors (Guy, §B4). Related progress bounds the coprimality that is achievable: Bratley–McKay conjectured no amicable pair coprime to 6, disproved by Battiato–Borho (1988); pairs coprime to 30 exist (Kohmoto, 1997, with 193-digit members), but a fully coprime ($\gcd=1$) pair has never been found. Extensive amicable-pair databases (Pedersen/Moews/Garcia) enumerate millions of pairs, none coprime. The tool: generation via Thabit/Borho-style rules and exhaustive $\sigma$-based search, with all $\sigma$ values checkable from factorizations. Open.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Amicable Pair — Wolfram MathWorld | website |
| REF-02 | Amicable numbers — Wikipedia | website |
Investigations · 0
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