"The integer 1 is a prime number."

mathematics · generated 2026-03-28 · v0.10.0

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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.

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.

Generated by proof-engine v0.10.0 on 2026-03-28.

counter-evidence search

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.
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.
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.

audit trail

Computation Traces ▸

A1: Positive divisors of 1: [1]
  A2: number of positive divisors of 1: len(divisors_of_1) = len([1]) = 1
  A3: is 1 > 1?: 1 > 1 = False
    1 > 1 is False
  A2: divisor count == 2?: 1 == 2 = False

Primary method: 1 is prime = False
  Fails criterion 1 (n > 1): 1 > 1 is False
  Fails criterion 2 (exactly 2 divisors): has 1 divisor(s), need 2

A4 cross-check (trial division): is_prime(1) = False
A5 cross-check (sympy.isprime): isprime(1) = False
  Claim evaluation: is 1 prime?: False == True = False

Source: proof.py inline output (execution trace)

Hardening Checklist ▸

Rule 1: N/A — pure computation, no empirical facts
Rule 2: N/A — pure computation, no empirical facts
Rule 3: date.today() used for generated_at timestamp
Rule 4: CLAIM_FORMAL with explicit operator_note documenting interpretation of "prime number" and criteria for disproof
Rule 5: Three adversarial checks investigated: historical definitions, modern authorities, and FTA implications. None break the proof.
Rule 6: N/A — pure computation, no empirical facts. Cross-checks use three mathematically independent methods: (1) definitional enumeration, (2) trial division algorithm, (3) sympy.isprime
Rule 7: compare() and explain_calc() imported from computations.py; no hard-coded constants
validate_proof.py result: PASS (14/14 checks passed, 0 issues, 0 warnings)

Source: author analysis

Generated by proof-engine v0.10.0 on 2026-03-28.

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