# Audit: 2026 being divisible by 2, 3, 7, 11, 13, 17 proves it has "hidden perfect number properties." - **Generated:** 2026-03-28 - **Reader summary:** [proof.md](proof.md) - **Proof script:** [proof.py](proof.py) --- ## Claim Specification | Field | Value | |-------|-------| | Subject | 2026 | | Property | SC1: number of claimed divisors {2,3,7,11,13,17} that actually divide 2026 == 6; SC2: 2026 has "hidden perfect number properties" (sum of proper divisors equals 2026) | | Operator | == | | Threshold | 6 (SC1: all 6 claimed divisors must hold) | | Operator Note | Compound claim requires both SC1 and SC2. SC1 is checked via modular arithmetic (n % d == 0) and GCD characterization (gcd(n,d) == d). "Hidden perfect number properties" is not a standard term — interpreted as: 2026 is a perfect number (σ_proper(n) == n). If SC1 is false, the compound claim is disproved regardless of SC2. | --- ## Fact Registry | ID | Label | Type | |----|-------|------| | A1 | 2026 divisible by 2: 2026 % 2 == 0 | Type A (computed) | | A2 | 2026 divisible by 3: 2026 % 3 == 0 | Type A (computed) | | A3 | 2026 divisible by 7: 2026 % 7 == 0 | Type A (computed) | | A4 | 2026 divisible by 11: 2026 % 11 == 0 | Type A (computed) | | A5 | 2026 divisible by 13: 2026 % 13 == 0 | Type A (computed) | | A6 | 2026 divisible by 17: 2026 % 17 == 0 | Type A (computed) | | A7 | Prime factorization of 2026 (trial division) | Type A (computed) | | A8 | SC2 check: sum of proper divisors of 2026 vs 2026 (perfect number test) | Type A (computed) | | A9 | Cross-check: divisibility via gcd(2026, d) == d for each claimed d | Type A (computed) | --- ## Full Evidence Table ### Type A (Computed) Facts | ID | Fact | Method | Result | |----|------|--------|--------| | A1 | 2026 divisible by 2 | 2026 % 2 | remainder=0, divisible=True | | A2 | 2026 divisible by 3 | 2026 % 3 | remainder=1, divisible=False | | A3 | 2026 divisible by 7 | 2026 % 7 | remainder=3, divisible=False | | A4 | 2026 divisible by 11 | 2026 % 11 | remainder=2, divisible=False | | A5 | 2026 divisible by 13 | 2026 % 13 | remainder=11, divisible=False | | A6 | 2026 divisible by 17 | 2026 % 17 | remainder=3, divisible=False | | A7 | Prime factorization of 2026 | Trial division prime factorization | 2 × 1013 | | A8 | SC2: perfect number test | Sum of proper divisors via trial division | sigma_proper(2026)=1016, required=2026, difference=−1010, classification=deficient | | A9 | Cross-check: GCD method | gcd(2026, d) == d for each d in {2,3,7,11,13,17} | GCD method: 1/6 divisibilities hold, agrees with modulo method (1/6) | *(No Type B empirical facts — pure-math proof.)* --- ## Computation Traces ``` === SC1: Divisibility of 2026 by claimed divisors [2, 3, 7, 11, 13, 17] === 2026 % 2 = 0 → divisible 2026 % 3 = 1 → NOT divisible (remainder=1) 2026 % 7 = 3 → NOT divisible (remainder=3) 2026 % 11 = 2 → NOT divisible (remainder=2) 2026 % 13 = 11 → NOT divisible (remainder=11) 2026 % 17 = 3 → NOT divisible (remainder=3) Divisors that hold: [2] Divisors that FAIL: [3, 7, 11, 13, 17] Count of divisibilities holding: 1 / 6 === Cross-Check: GCD method — gcd(2026, d) == d iff d | 2026 === gcd(2026, 2) = 2 → divisible gcd(2026, 3) = 1 → NOT divisible (gcd=1≠3) gcd(2026, 7) = 1 → NOT divisible (gcd=1≠7) gcd(2026, 11) = 1 → NOT divisible (gcd=1≠11) gcd(2026, 13) = 1 → NOT divisible (gcd=1≠13) gcd(2026, 17) = 1 → NOT divisible (gcd=1≠17) GCD method count of divisibilities holding: 1 / 6 Cross-check: modulo method and GCD method agree (1 == 1) ✓ === Prime Factorization of 2026 === 2026 = 2 × 1013 Verification: 2 × 1013 = 2026 ✓ Prime factors of 2026: [2, 1013] 3 in prime factors of 2026: False → 2026 is NOT divisible by 3 7 in prime factors of 2026: False → 2026 is NOT divisible by 7 11 in prime factors of 2026: False → 2026 is NOT divisible by 11 13 in prime factors of 2026: False → 2026 is NOT divisible by 13 17 in prime factors of 2026: False → 2026 is NOT divisible by 17 LCM of {2,3,7,11,13,17} = 2×3×7×11×13×17: a * b * c * d_val * e * f_val = 2 * 3 * 7 * 11 * 13 * 17 = 102102 The smallest positive number divisible by all 6 primes is 102102 2026 is 100076 less than 102102 — it is NOT a multiple of 102102 === SC2: Perfect Number Test for 2026 === Proper divisors of 2026: [1, 2, 1013] Sum of proper divisors (sigma(n) - n): 1016 For a perfect number, sum must equal 2026 1016 == 2026: False Difference: 1016 - 2026 = -1010 Classification: deficient Euclid-Euler test: does 2026 = 2^(p-1) * (2^p - 1) for any prime p? No p in [2,19] satisfies 2^(p-1) × (2^p - 1) = 2026 Euclid-Euler sequence (p=2,3,5,7): [6, 28, 496, 8128] 2026 does not appear in this sequence sigma(2026) = 3042 (sum of ALL divisors including 2026) sigma(2026) / 2026 = 1.501481 (perfect numbers have ratio = 2.000000) SC1: count of divisibilities holding == 6 (all must hold): 1 == 6 = False SC2: sum of proper divisors == 2026 (perfect number test): 1016 == 2026 = False ``` --- ## Independent Cross-Check (Rule 6) | Description | Value A (Modulo) | Value B (GCD) | Agreement | |-------------|-----------------|---------------|-----------| | Count of divisibilities holding | 1 | 1 | True | **Method independence:** The modulo method tests `n % d == 0` directly. The GCD method uses the algebraic identity `d | n ⟺ gcd(n, d) = d`, which is a structurally different characterization (Euclidean algorithm vs. remainder computation). Both methods yield the same count (1/6), confirming the result is not an artifact of any single method. --- ## Adversarial Checks (Rule 5) **Check 1: Could '2026' refer to a transformed value?** - Question: Could '2026' in the claim refer to a transformed or encoded value that IS divisible by all six primes? - Verification performed: Computed LCM(2,3,7,11,13,17) = 102102. Checked all multiples of 102102 near 2026: the nearest multiples are 0 and 102102. 2026 is not a multiple of 102102. No standard calendar year encoding (e.g., 2026 mod k, 2026 in a different base) produces 102102 or any multiple thereof. Checked 2026 in bases 2–16: none yield 102102. - Finding: 2026 in any standard encoding is not divisible by all six claimed primes. The premise is straightforwardly false for the integer 2026. - Breaks proof: No **Check 2: Is "hidden perfect number properties" a recognized term?** - Question: Is 'hidden perfect number properties' a recognized mathematical term that 2026 could satisfy? - Verification performed: Surveyed standard number theory classifications: perfect numbers (sigma(n)=2n), quasi-perfect (sigma(n)=2n+1, none known), almost perfect (sigma(n)=2n−1, only powers of 2), multiply perfect/k-perfect (sigma(n)=kn), semiperfect/pseudoperfect (n equals some subset-sum of proper divisors), weird numbers (abundant but not semiperfect). "Hidden perfect number properties" appears in none of these standard classifications. Searched for the phrase in number theory literature — no results found. - Finding: 'Hidden perfect number properties' has no standard mathematical definition. Under the most charitable interpretation (2026 is a perfect number): sum of proper divisors of 2026 = 1016 ≠ 2026. 2026 is a deficient number. sigma(2026)/2026 ≈ 1.501, far from the ratio of 2 required for a perfect number. - Breaks proof: No **Check 3: Does divisibility by {2,3,7,11,13,17} imply perfect-number-adjacent properties?** - Question: Does divisibility by 2, 3, 7, 11, 13, 17 imply any known perfect-number-adjacent property for numbers that ARE divisible by all six? - Verification performed: The smallest number divisible by 2,3,7,11,13,17 is their LCM = 102102. For squarefree n = p₁×…×p_k, sigma(n) = (1+p₁)(1+p₂)…(1+p_k). sigma(102102) = 3×4×8×12×14×18 = 290304. sigma(102102)/102102 ≈ 2.843 ≠ 2. So 102102 is abundant, not perfect. - Finding: Even the smallest number genuinely divisible by all six claimed primes (102102) is NOT a perfect number. Divisibility by {2,3,7,11,13,17} does not imply perfect number properties for any number, let alone for 2026 which fails the divisibility premise. - Breaks proof: No --- *Generated by [proof-engine](https://github.com/yaniv-golan/proof-engine) v0.10.0 on 2026-03-28.*