# Proof: "Thank God 2026 has no square root" is mathematically meaningful

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

---

## Key Findings

- **2026 is not a perfect square:** ⌊√2026⌋ = 45, and 45² = 2025 ≠ 2026, while 46² = 2116 ≠ 2026. No integer squares to 2026 (confirmed by exhaustive search over all n ∈ [1, 2026]).
- **Two independent methods agree:** The integer square root (isqrt) method and the prime factorisation method both independently confirm 2026 is not a perfect square.
- **2026 = 2¹ × 1013¹** — both prime factors appear to an *odd* exponent, which by the fundamental theorem of arithmetic is a necessary and sufficient condition for not being a perfect square.
- **Context makes the statement meaningful:** The immediately preceding year, 2025 = 45², *is* a perfect square. The "Thank God" relief is mathematically grounded: one more year and the perfect-square property vanishes.
- **Bonus:** √2026 is irrational — proved via a 2-adic valuation argument.

---

## Claim Interpretation

**Natural claim:** "Thank God 2026 has no square root" is mathematically meaningful.

**Formal interpretation:** The phrase "has no square root" is a colloquial shorthand for "is not a perfect square" — i.e., no positive integer n satisfies n² = 2026. The claim is *mathematically meaningful* if and only if it makes a precise, true mathematical assertion. We formalise this as:

> **Count of positive integers n with n² = 2026 = 0**

The claim would be false — and thus not supported — if any n ∈ ℤ⁺ existed with n² = 2026.

**Operator choice:** The `==` operator with threshold 0 is the natural formalisation: we assert the *count* of integer square roots is exactly zero, which is a stronger statement than `< 1` and makes the structure of the claim explicit.

**Important note on real square roots:** Every positive real number has a real-valued square root. √2026 ≈ 45.011… exists in ℝ. The mathematically interesting (and true) assertion is about *integer* square roots. This is consistent with the colloquial usage in year-numbering contexts (e.g., "2025 is a perfect square year").

---

## Evidence Summary

| ID | Fact | Verified |
|----|------|---------|
| A1 | isqrt(2026)² ≠ 2026 (primary: no integer square root) | Computed: True — 45² = 2025 ≠ 2026 and 46² = 2116 ≠ 2026 |
| A2 | Prime factorisation of 2026 has odd-exponent prime factors (cross-check) | Computed: True — 2026 = 2¹ × 1013¹, both exponents odd |
| A3 | 45² = 2025 (predecessor year is a perfect square — context) | Computed: True — 2025 = 45² confirmed |
| A4 | √2026 is irrational | Computed: True — valuation argument (see Proof Logic) |

---

## Proof Logic

### Sub-claim 1 (A1): No integer square root — isqrt method

Python's `math.isqrt(n)` returns the largest integer k such that k² ≤ n. For n = 2026:

```
math.isqrt(2026) = 45
45² = 2025 ≠ 2026   (one short)
46² = 2116 ≠ 2026   (already past)
```

Since 2026 lies strictly between 45² = 2025 and 46² = 2116, there is no integer whose square equals 2026 (A1).

### Sub-claim 2 (A2): No integer square root — prime factorisation method

A positive integer is a perfect square if and only if every prime in its factorisation appears to an even exponent (Fundamental Theorem of Arithmetic). Factorising 2026:

```
2026 = 2¹ × 1013¹
```

Both 2 and 1013 are prime (1013's primality verified by trial division up to ⌊√1013⌋ = 31). Both exponents are 1 — odd. Therefore 2026 is not a perfect square (A2).

**Cross-check (A1 vs A2):** The two methods use completely different mathematical machinery (integer square root floor/ceiling vs prime factorisation and exponent parity) and agree: 2026 is not a perfect square.

### Sub-claim 3 (A3): 2025 is a perfect square — context

```
math.isqrt(2025) = 45
45² = 2025 ✓
```

The year 2025 is a perfect square (A3). This gives the colloquial claim its meaning: the property "is a perfect square" held for 2025 and does not hold for 2026. The relief expressed by "Thank God" is mathematically grounded.

### Sub-claim 4 (A4): √2026 is irrational

**Proof by contradiction.** Suppose √2026 = p/q in lowest terms (p, q ∈ ℤ⁺, gcd(p,q) = 1). Then p² = 2026 q².

Let v₂(m) denote the 2-adic valuation of m (exponent of 2 in the prime factorisation of m). Since 2026 = 2¹ × 1013¹:

- Left side: v₂(p²) = 2·v₂(p) — always **even**
- Right side: v₂(2026 q²) = v₂(2026) + v₂(q²) = 1 + 2·v₂(q) — always **odd**

Even = Odd is a contradiction. Therefore √2026 is irrational (A4).

---

## Counter-Evidence Search

Four adversarial questions were investigated before writing the proof:

1. **"Does 2026 have an integer square root that was overlooked?"**
   Exhaustive computation: for all n ∈ [1, 2026], n² ≠ 2026. Zero counterexamples found.

2. **"Could 'no square root' mean something other than 'not a perfect square'?"**
   Three interpretations were examined: (a) no real square root — FALSE (√2026 ≈ 45.011 exists), (b) no rational square root — TRUE, (c) no integer square root — TRUE. Only interpretation (a) would falsify the claim, but it is inconsistent with the celebratory "Thank God" framing used in the context of perfect-square years. The claim is unambiguously meaningful.

3. **"Is 2026 perhaps a perfect power of another kind?"**
   Since 2026 = 2 × 1013 (semiprimes with exponents all equal to 1), it cannot be a perfect k-th power for any k ≥ 2. The claim is specific to squares and is not undermined by other power relationships.

4. **"Is 1013 actually prime?"**
   Trial division by all primes up to ⌊√1013⌋ = 31 found no divisors. 1013 is prime.

No adversarial check breaks the proof.

---

## Conclusion

**PROVED.** The statement "Thank God 2026 has no square root" is mathematically meaningful and correct under the natural interpretation: 2026 is not a perfect square. This was established by two independent computational methods:

- **isqrt method (A1):** 45² = 2025 < 2026 < 2116 = 46² — no integer squares to 2026.
- **Factorisation method (A2):** 2026 = 2¹ × 1013¹ — odd exponents preclude perfect-square status.

Both methods agree. Additionally, √2026 is irrational (A4), and the immediately preceding year 2025 = 45² *is* a perfect square (A3), grounding the "relief" expressed in the original claim. The exhaustive adversarial search found zero counterexamples.

All results are pure Type A (computed) facts — no citations, no external sources, fully reproducible by running `python proof.py`.

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