# Proof: 2026 has no square root

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

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## Key Findings

- The nearest perfect squares bracket 2026: **45² = 2025** and **46² = 2116**, with no integer between 45 and 46 — so no integer n satisfies n² = 2026.
- Independent confirmation via modular arithmetic: perfect squares mod 16 can only be 0, 1, 4, or 9. **2026 mod 16 = 10**, which is not in that set, proving 2026 cannot be a perfect square.
- Both methods agree: 2026 has **no integer square root**.
- Context: **2025 = 45²** is a perfect square, making the year 2025 special — and 2026 immediately follows it without inheriting that property.

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## Claim Interpretation

**Natural language claim:** "2026 has no square root"

The claim is interpreted as: *there is no integer n ≥ 0 satisfying n² = 2026* (equivalently, 2026 is not a perfect square).

In ℝ, 2026 trivially has square roots: √2026 ≈ 45.011…, which is irrational but perfectly real. In ℂ, square roots always exist. Under those interpretations the claim would be **false**, which makes it uninteresting.

The viral meme's mathematical punch-line — that 2025 = 45² is a perfect square year and 2026 is not — only makes sense under the **integer interpretation**. This is the standard meaning of "has a square root" in elementary number theory (i.e., is a perfect square). The formal claim is scoped accordingly.

Formally:

| Field | Value |
|-------|-------|
| Subject | 2026 |
| Property | existence of an integer n such that n² = 2026 |
| Operator | == |
| Threshold | False (such n does NOT exist) |

Proof succeeds when both independent methods agree that no such integer exists.

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## Evidence Summary

| ID | Fact | Verified |
|----|------|---------|
| A1 | Floor-sqrt bound check: no integer squares to 2026 | Computed: 45² = 2025 ≠ 2026, 46² = 2116 ≠ 2026 → no integer square root |
| A2 | Quadratic-residue mod-16 check: 2026 mod 16 is not a perfect-square residue | Computed: 2026 mod 16 = 10; QR₁₆ = {0,1,4,9}; 10 ∉ QR₁₆ → not a perfect square |
| A3 | Context: 2025 = 45² confirming the adjacent perfect square | Computed: 45² = 2025 → 2025 IS a perfect square (adjacent year) |

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## Proof Logic

**Method A1 — Floor-sqrt bound check**

Python's `math.isqrt(n)` returns the exact integer part of √n (no floating-point rounding; arbitrary precision for all non-negative integers). For n = 2026:

- `math.isqrt(2026)` = **45**
- 45² = **2025** (A1) — one less than 2026
- 46² = **2116** (A1) — 90 more than 2026

Since 2025 < 2026 < 2116, the number 2026 lies strictly between two consecutive perfect squares. There is no integer between 45 and 46, so there is no integer whose square equals 2026.

**Method A2 — Quadratic residues mod 16 (independent cross-check)**

A fundamental result in modular arithmetic: if n is a perfect square, then n mod 16 must be one of **{0, 1, 4, 9}** (these are all possible values of k² mod 16 for integer k, verified by exhaustive enumeration of k = 0…15).

2026 mod 16 = **10** (A2). Since 10 ∉ {0, 1, 4, 9}, 2026 is provably not a perfect square by this method alone — without computing any square roots.

Both methods return the same result: is_perfect_square = **False**. They share no intermediate computation (A1 uses integer square roots; A2 uses only modular reduction), making their agreement a genuine independent confirmation.

**A3 — Why 2025 is special**

For completeness: `math.isqrt(2025)` = 45, and 45² = 2025 exactly. So **2025 is a perfect square** (A3) — the last one before 2026. This is the basis of the viral meme: the year 2025 enjoyed the rare distinction of being a perfect square (45²), and 2026 does not.

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## Counter-Evidence Search

Four potential challenges were investigated:

1. **Real/complex square roots**: √2026 ≈ 45.011… does exist in ℝ. Under the real interpretation the claim is false. However, the meme's entire mathematical point requires the integer interpretation; the real case is explicitly acknowledged and excluded from the formal claim.

2. **Neighbouring perfect squares**: 44² = 1936, 45² = 2025, 46² = 2116, 47² = 2209. 2026 falls strictly between 2025 and 2116 — no adjacent perfect square was missed.

3. **Potential modular-arithmetic error**: The residue set {0,1,4,9} was verified by enumerating all k = 0…15. The value 2026 mod 16 = 10 was confirmed by direct subtraction (2026 − 126×16 = 2026 − 2016 = 10). Both are verifiable by hand.

4. **Reliability of math.isqrt()**: The function is exact (PEP 578, Python 3.8+) with no floating-point rounding risk. The result for 2026 is independently bounded by the bracketing perfect squares.

None of these break the proof.

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## Conclusion

**Verdict: PROVED**

There is no integer n satisfying n² = 2026. This is confirmed by two independent methods:
- **A1 (floor-sqrt):** The nearest integer square root of 2026 is 45, but 45² = 2025 ≠ 2026.
- **A2 (mod 16):** 2026 mod 16 = 10, which is not a quadratic residue mod 16 — a sufficient condition for non-perfect-squareness.

The proof is entirely computational (Type A), requires no external sources, and can be re-run by anyone with Python 3.8+.

**The meme's mathematical punchline is correct**: 2025 = 45² was the last perfect square year, and 2026 has no integer square root.

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