The Associated Pell GCD Entry-Point Conjecture

Abstract

The value k = 12 refutes the printed associated Pell gcd-entry-point biconditional.

Definition 1.1 (Mbirika-Schrader-Spilker Conjecture 32).

$$(claim)\iff (\forall k\in \mathrm{Nat},(1\le k)\Rightarrow ((1<\gcd \left(Q\left(k\right),k\right))\iff (\exists p\in \mathrm{Nat},(\mathrm{Prime}\left(p\right))\wedge \left((p\mid k)\wedge (\exists r\in \mathrm{Nat},(0<r)\wedge \left((p\mid Q\left(r\right))\wedge \left((\forall s\in \mathrm{Nat},(0<s)\Rightarrow ((s<r)\Rightarrow (\neg (p\mid Q\left(s\right)))))\wedge (r\mid k)\right)\right))\right))))$$

Formalization. D5/S1/Recurrence/Invariants/MbirikaAssociatedPellGcdEntryPointRefutation.claim (✓ std3).

Citation. aBa Mbirika; Janee Schrader; Jürgen Spilker (2023). Pell and Associated Pell Braid Sequences as GCDs of Sums of k Consecutive Pell, Balancing, and Related Numbers. DOI: 10.48550/arXiv.2301.05758. URL: https://cs.uwaterloo.ca/journals/JIS/VOL26/Mbirika/mbir5.pdf.

Commentary.

Conjecture 32 states verbatim: “We claim that gcd(Q_k, k) > 1 if and only if there exists a prime p such that p divides k and the rank of apparition (or entry point), e_Q(p) divides k. For example, gcd(Q_21, 21) = 7 and for the prime p = 7, we have p divides 21 and e_Q(p) = 3 divides 21.” The symbol Q is the associated Pell sequence already defined in D5/S1/Recurrence/PellCompanionGcd, with initial values one and one. The existential r expresses that the least positive index where p divides Q exists, and that this index divides k.

Theorem 1.2 (The conjecture fails at k = 12).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Invariants/MbirikaAssociatedPellGcdEntryPointRefutation.result (✓ std3). ∎

Resolves. Problems/mbirika-schrader-spilker-associated-pell-gcd-entry-point-refutation (refuted) by D5/S1/Recurrence/Invariants/MbirikaAssociatedPellGcdEntryPointRefutation.result.

Source. Repository-derived.

Acknowledgement. aBa Mbirika; Janee Schrader; Jürgen Spilker (2023). Pell and Associated Pell Braid Sequences as GCDs of Sums of k Consecutive Pell, Balancing, and Related Numbers. DOI: 10.48550/arXiv.2301.05758. URL: https://cs.uwaterloo.ca/journals/JIS/VOL26/Mbirika/mbir5.pdf.

Commentary.

At k = 12, Q(12) = 19601 and gcd(Q(12),12) = 1, so the left side is false. The prime p = 3 divides 12, its least positive entry point is r = 2 because Q(1) = 1 and Q(2) = 3, and 2 divides 12. Thus the right side is true and the biconditional is false.

References

• Truth anchor: D5/S1/Recurrence/Invariants/MbirikaAssociatedPellGcdEntryPointRefutation.claim
• Truth anchor: D5/S1/Recurrence/Invariants/MbirikaAssociatedPellGcdEntryPointRefutation.result
• Dependency: D5/S1/Recurrence/PellCompanionGcd