# Proof: The 100000th prime number is exactly 1299709

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

## Evidence Summary

| ID | Fact | Verified |
|----|------|----------|
| A1 | 100000th prime via Sieve of Eratosthenes | Computed: 1,299,709 |
| A2 | Prime-counting function pi(1299709) equals 100000 | Computed: 100,000 primes found |
| A3 | 1299709 is prime (trial division) | Computed: True |
| A4 | pi(1299708) equals 99999 (confirms no prime between) | Computed: 99,999 primes found |

*Source: proof.py JSON summary*

## Proof Logic

The proof uses three mathematically independent methods that together establish the claim:

**Primary computation (A1):** A Sieve of Eratosthenes generates all primes up to a computed upper bound and extracts the 100,000th prime. The result is 1,299,709.

**Cross-check via counting (A2, A4):** An independent sieve implementation counts all primes up to 1,299,709, finding exactly 100,000 (A2). Counting primes up to 1,299,708 yields 99,999 (A4). Together, these confirm that 1,299,709 is prime and is the exact 100,000th prime — not the 99,999th or 100,001st.

**Cross-check via trial division (A3):** A trial division algorithm independently confirms that 1,299,709 has no divisors other than 1 and itself, verifying its primality without relying on any sieve.

**Small-case validation:** The sieve was verified against known values: p(1)=2, p(10)=29, p(100)=541, p(1000)=7919 — all correct, ruling out systematic off-by-one errors.

*Source: author analysis*

## What could challenge this verdict?

Three adversarial checks were performed:

1. **Indexing convention:** Verified that the standard convention p(1)=2 is used consistently. A 0-based convention would shift the answer, but both the claim and the computation use 1-based indexing.

2. **Historical treatment of 1 as prime:** If 1 were counted as prime, the index would shift by one. Modern mathematical convention (post-1800s) excludes 1, and the proof follows this convention.

3. **Off-by-one errors:** The sieve was validated against four known small primes. The boundary check (pi(1299709)=100000 and pi(1299708)=99999) structurally rules out off-by-one errors in the primary computation.

No counter-evidence was found.

*Source: proof.py JSON summary*

## Conclusion

**PROVED.** The 100,000th prime number is exactly 1,299,709. This was established by three independent computational methods: a Sieve of Eratosthenes finding the nth prime (A1), a prime-counting function confirming pi(1,299,709) = 100,000 (A2) and pi(1,299,708) = 99,999 (A4), and trial division confirming 1,299,709 is prime (A3). All methods agree. The proof is purely computational with no empirical citations required.