The OEIS A100952 Log-Bounded Semiprime-Partition Conjecture
Abstract
A residue-class obstruction refutes the log-bounded partition conjecture in A100952.
Definition 1.1 (A log-bounded prime-plus-semiprime representation).
Formalization. D5/S3/Arith/Congruence/VosPostLogBoundedSemiprimePartitionRefutation.Rep (✓ std3).
Citation. Jonathan Vos Post; Reinhard Zumkeller (2004). OEIS A100952, Numbers that cannot be written as pq+r with three distinct primes p, q and r*. URL: https://oeis.org/A100952.
Commentary.
The three variables p, q, and r are natural primes. The inequality casts p and min(q,r) to real numbers and uses Real.log, the natural logarithm. Equal semiprime factors q and r are allowed.
Definition 1.2 (The eventual log-bounded partition conjecture).
Formalization. D5/S3/Arith/Congruence/VosPostLogBoundedSemiprimePartitionRefutation.claim (✓ std3).
Citation. Jonathan Vos Post; Reinhard Zumkeller (2004). OEIS A100952, Numbers that cannot be written as pq+r with three distinct primes p, q and r*. URL: https://oeis.org/A100952.
Commentary.
This is the source’s allowance for a threshold larger than sixty: some natural B at least sixty works for every natural m above B.
Theorem 1.3 (The eventual conjecture is false).
Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/VosPostLogBoundedSemiprimePartitionRefutation.result (✓ std3). ∎
Resolves. Problems/oeis-a100952-log-bounded-semiprime-partition-refutation (refuted) by D5/S3/Arith/Congruence/VosPostLogBoundedSemiprimePartitionRefutation.result.
Source. Repository-derived.
Acknowledgement. Jonathan Vos Post; Reinhard Zumkeller (2004). OEIS A100952, Numbers that cannot be written as pq+r with three distinct primes p, q and r*. URL: https://oeis.org/A100952.
Commentary.
Every number 6t+5 fails. The logarithmic estimate first forces q and r above three; parity then forces p=2; divisibility by three finally forces q=3 or r=3, a contradiction. The completeness conjecture for A100952 is untouched.