# Proof: The integer 1 is a prime number.

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

## Key Findings

- The integer 1 has only **one** positive divisor (itself), but a prime number must have exactly **two** distinct positive divisors (1 and itself).
- The integer 1 fails the "greater than 1" criterion required by the standard definition of prime numbers.
- Three independent methods (definitional check, trial division, sympy) all confirm 1 is **not** prime.
- The exclusion of 1 from the primes is universal in modern mathematics and is necessary to preserve the Fundamental Theorem of Arithmetic.

## Claim Interpretation

**Natural language:** "The integer 1 is a prime number."

**Formal interpretation:** A prime number is defined as a natural number greater than 1 whose only positive divisors are 1 and itself. For 1 to be prime, it must satisfy both conditions: (1) be greater than 1, and (2) have exactly two distinct positive divisors. The claim asserts that 1 is prime; disproof requires showing it fails at least one definitional criterion.

## Evidence Summary

| ID | Fact | Verified |
|----|------|----------|
| A1 | Positive divisors of 1 (exhaustive enumeration) | Computed: [1] — only divisor is 1 itself |
| A2 | Count of positive divisors of 1 | Computed: 1 (a prime requires exactly 2) |
| A3 | Whether 1 > 1 (greater-than-1 criterion) | Computed: False |
| A4 | Cross-check: trial division primality test | Computed: False (1 is not prime) |
| A5 | Cross-check: sympy.isprime(1) | Computed: False (1 is not prime) |

## Proof Logic

The proof applies the standard definition of prime numbers directly to the integer 1.

**Step 1 — Enumerate divisors:** All positive divisors of 1 are computed by exhaustive enumeration over [1, n]. The only divisor is 1 itself (A1). This gives a divisor count of 1 (A2), but a prime must have exactly 2 distinct positive divisors (1 and the number itself). Since 1 has only 1 divisor, it fails this criterion.

**Step 2 — Check greater-than-1 criterion:** The definition requires primes to be strictly greater than 1. Since 1 > 1 is false (A3), this criterion also fails.

**Step 3 — Cross-checks:** Two independent methods confirm the result:
- A standard trial division primality test returns False for 1 (A4).
- Python's sympy library, an independent mathematical computation engine, confirms `isprime(1) = False` (A5).

The integer 1 fails **both** definitional criteria for primality, and all three independent methods agree unanimously.

## Counter-Evidence Search

1. **Was 1 ever historically considered prime?** Yes — until the mid-19th century, mathematicians including Goldbach, Euler, and Lebesgue sometimes considered 1 to be prime. However, the modern convention (universally adopted since ~1899) excludes 1, and this proof evaluates the claim against the current standard definition.

2. **Is there any modern mathematical authority that defines 1 as prime?** No. ISO 80000-2, major textbooks (Hardy & Wright, Niven Zuckerman & Montgomery, Ireland & Rosen), and computational references (OEIS A000040) all define primes as integers greater than 1.

3. **Does the Fundamental Theorem of Arithmetic break if 1 is prime?** Yes. If 1 were prime, factorizations would not be unique (e.g., 6 = 2 x 3 = 1 x 2 x 3 = 1 x 1 x 2 x 3). This is the key mathematical reason 1 is excluded — the convention is not arbitrary but mathematically necessary.

## Conclusion

**DISPROVED.** The integer 1 is not a prime number. It fails both criteria of the standard definition: (1) it is not greater than 1, and (2) it has only 1 positive divisor rather than the required 2. Three independent computational methods (definitional check, trial division, sympy) unanimously confirm that 1 is not prime. While 1 was historically sometimes considered prime, the modern mathematical consensus — universally adopted for over a century — excludes it to preserve the Fundamental Theorem of Arithmetic.

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