# Proof: 2026 being divisible by 2, 3, 7, 11, 13, 17 proves it has "hidden perfect number properties."

- **Generated:** 2026-03-28
- **Verdict:** DISPROVED
- **Audit trail:** [proof_audit.md](proof_audit.md) | [proof.py](proof.py)

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## Key Findings

- **The premise is false:** 2026 is divisible by only 1 of the 6 claimed primes. It is divisible by 2, but NOT by 3, 7, 11, 13, or 17. Confirmed by two independent methods (modular arithmetic and GCD characterization), both agreeing.
- **2026 = 2 × 1013**, where 1013 is prime. It has exactly three divisors: 1, 2, and 1013.
- **2026 is not a perfect number:** Its proper divisors (1, 2, 1013) sum to 1016 — a deficit of 1010 from the 2026 required. It is classified as a *deficient* number, with sigma(n)/n ≈ 1.501 vs. the ratio of 2.000 required for perfection.
- **"Hidden perfect number properties" has no mathematical definition.** Under the most charitable interpretation (2026 is a perfect number), the claim is also false independently of the divisibility premise.

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## Claim Interpretation

**Natural language claim:** 2026 being divisible by 2, 3, 7, 11, 13, 17 proves it has "hidden perfect number properties."

**Formal interpretation:** This is a compound claim with two sub-claims:

- **SC1 (Premise):** 2026 is divisible by each of 2, 3, 7, 11, 13, and 17 — i.e., all 6 of these integers divide 2026 with remainder 0.
- **SC2 (Conclusion):** 2026 has "hidden perfect number properties."

For the compound claim to be true, SC1 must hold (the divisibility premise must be correct) AND SC2 must be meaningful and true.

**Operator choice:** The threshold is 6 — all 6 claimed divisors must hold. If even one fails, SC1 is false. In practice, 5 of 6 fail, making the premise comprehensively false.

**"Hidden perfect number properties"** is not a recognized term in number theory. We adopt the most charitable interpretation: either (a) 2026 is itself a perfect number (i.e., the sum of its proper divisors equals 2026), or (b) 2026 fits the Euclid-Euler formula for even perfect numbers (2^(p−1) × (2^p − 1) for Mersenne prime p). A false premise cannot logically "prove" anything, but we evaluate SC2 independently for completeness.

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## Evidence Summary

| ID | Fact | Verified |
|----|------|---------|
| A1 | 2026 divisible by 2: 2026 % 2 == 0 | Computed: remainder=0, divisible=True |
| A2 | 2026 divisible by 3: 2026 % 3 == 0 | Computed: remainder=1, divisible=**False** |
| A3 | 2026 divisible by 7: 2026 % 7 == 0 | Computed: remainder=3, divisible=**False** |
| A4 | 2026 divisible by 11: 2026 % 11 == 0 | Computed: remainder=2, divisible=**False** |
| A5 | 2026 divisible by 13: 2026 % 13 == 0 | Computed: remainder=11, divisible=**False** |
| A6 | 2026 divisible by 17: 2026 % 17 == 0 | Computed: remainder=3, divisible=**False** |
| A7 | Prime factorization of 2026 | Computed: 2 × 1013 |
| A8 | SC2: sum of proper divisors vs 2026 (perfect number test) | Computed: sigma_proper(2026)=1016, required=2026, difference=−1010, deficient |
| A9 | Cross-check: GCD method agrees with modulo method | Computed: 1/6 divisibilities hold by both methods |

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## Proof Logic

**SC1 — Divisibility premise (A1–A6, A9):**

Each divisibility claim was checked by computing 2026 % d for each claimed divisor d (A1–A6). Only 2026 % 2 = 0 — 2026 is even. The remaining five divisors all fail with nonzero remainders: 2026 % 3 = 1, 2026 % 7 = 3, 2026 % 11 = 2, 2026 % 13 = 11, 2026 % 17 = 3 (A2–A6).

An independent cross-check using the GCD characterization (d | n ⟺ gcd(n, d) = d) confirms all five failures identically (A9): gcd(2026, 3) = gcd(2026, 7) = gcd(2026, 11) = gcd(2026, 13) = gcd(2026, 17) = 1, not equal to d in any case.

The prime factorization (A7) explains why: **2026 = 2 × 1013**, where 1013 is prime. None of 3, 7, 11, 13, 17 appear as factors, so 2026 cannot be divisible by any of them.

For comparison, the smallest positive integer divisible by all six of these primes is their LCM = 2 × 3 × 7 × 11 × 13 × 17 = **102,102** — a number 100,076 larger than 2026.

**SC2 — Perfect number test (A8):**

A perfect number is formally defined as a positive integer n such that the sum of its proper divisors equals n. The proper divisors of 2026 are {1, 2, 1013}, which sum to 1016 — falling short of 2026 by 1010 (A8). The ratio σ(2026)/2026 ≈ 1.501, compared to the ratio of exactly 2.000 required for a perfect number. 2026 is therefore classified as *deficient*.

The Euclid-Euler formula for even perfect numbers (2^(p−1) × (2^p − 1) for Mersenne prime p) generates the sequence 6, 28, 496, 8128, … — 2026 does not appear in this sequence.

**Compound verdict:** SC1 is false (the premise is wrong), and SC2 is independently false. The compound claim fails on both counts.

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## Counter-Evidence Search

**Could 2026 refer to a transformed value?** The LCM of {2,3,7,11,13,17} is 102,102. No standard encoding of the calendar year 2026 (different bases, modular transforms) produces 102,102 or any multiple thereof. The premise is false for 2026 under any standard interpretation.

**Is "hidden perfect number properties" a recognized term?** Standard number theory encompasses perfect, quasi-perfect, almost perfect, multiply perfect (k-perfect), semiperfect/pseudoperfect, and weird numbers. "Hidden perfect number properties" appears in none of these classifications and was not found in a search of number theory literature. The term appears to be invented.

**Does divisibility by {2,3,7,11,13,17} imply perfect-number-adjacent properties?** Even if a number were actually divisible by all six primes, it would not thereby acquire perfect number properties. The smallest such number, 102,102 = 2×3×7×11×13×17, is *abundant* (σ(102102)/102102 ≈ 2.843 ≠ 2), not perfect. Divisibility by these six primes has no mathematical connection to perfect numbers.

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## Conclusion

**DISPROVED.** The claim has a false premise and a false conclusion:

- **SC1 is false:** 2026 is divisible by only 1 of the 6 claimed primes (by 2, not by 3, 7, 11, 13, or 17). Five independent divisibility checks fail, confirmed by two independent algebraic methods.
- **SC2 is independently false:** 2026 is a deficient number with σ_proper(2026) = 1016 ≠ 2026. It does not satisfy the definition of a perfect number under any standard interpretation, and "hidden perfect number properties" is not a recognized mathematical concept.

The proof depends entirely on arithmetic computations (Type A facts). There are no citations, no empirical sources, and no verification uncertainty. `python proof.py` reproduces all results deterministically.

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*Generated by [proof-engine](https://github.com/yaniv-golan/proof-engine) v0.10.0 on 2026-03-28.*