"Fewer than 20 percent of the positive integers from 1 to 1000 are prime."

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

⬡ Verified by Proof Engine — an open-source tool that proves claims using cited sources and executable code. No LLM trust required.
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Key Findings

There are exactly 168 prime numbers in the range [1, 1000], computed by two independent algorithms that agree perfectly.
168 out of 1000 is 16.8%, which is strictly less than 20%.
The prime count of 168 matches the well-known number-theoretic value pi(1000) = 168.
No adversarial interpretation of the claim changes the result.

Claim Interpretation

Natural language: Fewer than 20 percent of the positive integers from 1 to 1000 are prime.

Formal interpretation: The count of primes in {1, 2, ..., 1000} divided by 1000, expressed as a percentage, is strictly less than 20. Equivalently, the prime count must be strictly less than 200. The phrase "fewer than" unambiguously implies strict inequality (<).

evidence summary

ID	Fact	Verified
A1	Prime count via Sieve of Eratosthenes	Computed: 168 primes
A2	Prime count via trial division (independent cross-check)	Computed: 168 primes
A3	Percentage of primes in [1, 1000]	Computed: 16.8%

Proof Logic

Two independent algorithms enumerate all primes in [1, 1000]:

Sieve of Eratosthenes (A1): Marks composites by iterating through multiples of each prime up to sqrt(1000). Produces a list of 168 primes, from 2 to 997.
Trial division (A2): Tests each integer in [1, 1000] individually for primality by checking divisibility up to its square root. Also produces exactly 168 primes.

Both methods yield identical prime lists (A1, A2 — independently computed). The percentage is then 168 / 1000 x 100 = 16.8% (A3), which satisfies 16.8 < 20.

Conclusion

PROVED. There are exactly 168 prime numbers among the positive integers from 1 to 1000, which is 16.8% — strictly fewer than 20%. This was established by two independent computational methods (Sieve of Eratosthenes and trial division) that produce identical results, consistent with the known value pi(1000) = 168.

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

counter-evidence search

Could the algorithms have bugs that undercount primes? Two independent algorithms (sieve and trial division) produce identical results. The count of 168 matches the well-known value pi(1000) = 168 from number theory. No undercounting detected.
Could "20 percent" be interpreted differently? "Fewer than" is unambiguously strict inequality. Even under a non-strict reading (<=), 168 <= 200 holds. No interpretation changes the result.

audit trail

Computation Traces ▸

Sieve of Eratosthenes: found 168 primes in [1, 1000]
  First 10: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
  Last 10:  [937, 941, 947, 953, 967, 971, 977, 983, 991, 997]

Trial division: found 168 primes in [1, 1000]
Cross-check: both methods agree — 168 primes
  percentage of primes in [1, 1000]: prime_count_sieve / N * 100 = 168 / 1000 * 100 = 16.8000
  prime percentage < 20%: 16.8 < 20 = True

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 generation date
Rule 4: CLAIM_FORMAL with operator_note explicitly documents the strict inequality interpretation
Rule 5: Two adversarial checks performed — algorithm correctness and interpretation ambiguity
Rule 6: N/A — pure computation, no empirical facts. Cross-check uses mathematically independent method (sieve vs trial division)
Rule 7: explain_calc() and compare() from computations.py used for all computations
validate_proof.py result: PASS (14/14 checks passed, 0 issues, 0 warnings)

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

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