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

## Verdict

**Verdict: DISPROVED**

The claim fails on both counts: the premise is factually wrong, and the conclusion it was meant to support is independently false.

## What was claimed?

The claim asserts that 2026 is divisible by six specific numbers — 2, 3, 7, 11, 13, and 17 — and that this divisibility somehow proves 2026 has special "hidden perfect number properties." It's the kind of claim that might circulate as a fun mathematical curiosity about the year 2026, suggesting something numerologically significant lurks beneath the surface.

## What did we find?

The first thing to check is whether 2026 is actually divisible by those six numbers. It isn't — not even close. Of the six claimed divisors, only one holds: 2026 is divisible by 2 (it's an even number). Every other divisor fails. Divide 2026 by 3 and you get a remainder of 1. Divide by 7 and the remainder is 3. By 11, remainder 2. By 13, remainder 11. By 17, remainder 3. Five out of six claims are simply wrong.

The arithmetic was checked two independent ways — once by computing remainders directly, and once using a different algebraic method based on greatest common divisors. Both approaches agree: only 1 of the 6 claimed divisibilities holds.

To understand why, consider what 2026 actually factors into: it equals 2 times 1013, where 1013 is itself a prime number. That's it — two prime factors. There's no 3, no 7, no 11, no 13, no 17 hiding anywhere in 2026's structure. For comparison, the smallest number that genuinely is divisible by all six of those primes is 102,102 — a number roughly fifty times larger than 2026.

Setting the false premise aside, what about the conclusion — does 2026 have "hidden perfect number properties"? A perfect number is one where all its smaller divisors add up to exactly the number itself. The classic example is 6: its divisors 1, 2, and 3 sum to exactly 6. For 2026, the proper divisors are 1, 2, and 1013. They sum to 1016 — falling short of 2026 by 1,010. That's not close. Mathematicians classify 2026 as a "deficient" number, the opposite of abundant, and nowhere near perfect.

As for the phrase "hidden perfect number properties" itself — it doesn't exist in mathematics. No standard classification, no published theorem, no recognized concept uses this term. It appears to have been invented.

## What should you keep in mind?

The claim has two separate failures, either of which would be enough to disprove it. Even in the most charitable reading — where "hidden perfect number properties" is interpreted as generously as possible — 2026 still doesn't qualify. There is no interpretation under which the conclusion holds.

It's also worth noting that even if a number *were* divisible by all six of those primes, that still wouldn't make it a perfect number. The smallest number divisible by 2, 3, 7, 11, 13, and 17 is 102,102, and it's not perfect either — it's actually abundant. Divisibility by this particular set of primes has no mathematical connection to perfect numbers whatsoever.

This proof is purely arithmetic — no citations, no empirical data, no interpretation judgment calls. The calculations are deterministic and exact.

## How was this verified?

Each divisibility check was computed directly and confirmed by a second independent algebraic method, with the prime factorization of 2026 providing a third line of evidence. See [the structured proof report](proof.md) for the full evidence table and logic chain, [the full verification audit](proof_audit.md) for computation traces and adversarial checks, or [re-run the proof yourself](proof.py) to reproduce every result from scratch.