Block #416,931

This is the prime chain origin stored in the block header. It is a composite number (not prime itself) — it equals the first prime in the chain multiplied by a primorial. The origin anchors the entire chain to this specific block.

3.434 × 10⁹⁸(99-digit number)

34341368518754200509…75079338983463575299

Discovered Prime Numbers

Lower: 2^k × origin − 1 | Upper: 2^k × origin + 1

These are the actual prime numbers discovered by this block, computed using the verified Primecoin formula. Each number has been independently confirmed to pass the Fermat primality test. Use the FactorDB links to verify any number independently.

Level 0 — Twin Prime Pair (origin ± 1)

origin − 1

3.434 × 10⁹⁸(99-digit number)

34341368518754200509…75079338983463575299

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.434 × 10⁹⁸(99-digit number)

34341368518754200509…75079338983463575301

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: origin + 1 − origin − 1 = 2 (twin primes ✓)

Level 1 — Twin Prime Pair (2^1 × origin ± 1)

2^1 × origin − 1

6.868 × 10⁹⁸(99-digit number)

68682737037508401018…50158677966927150599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.868 × 10⁹⁸(99-digit number)

68682737037508401018…50158677966927150601

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^1 × origin + 1 − 2^1 × origin − 1 = 2 (twin primes ✓)

Level 2 — Twin Prime Pair (2^2 × origin ± 1)

2^2 × origin − 1

1.373 × 10⁹⁹(100-digit number)

13736547407501680203…00317355933854301199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.373 × 10⁹⁹(100-digit number)

13736547407501680203…00317355933854301201

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^2 × origin + 1 − 2^2 × origin − 1 = 2 (twin primes ✓)

Level 3 — Twin Prime Pair (2^3 × origin ± 1)

2^3 × origin − 1

2.747 × 10⁹⁹(100-digit number)

27473094815003360407…00634711867708602399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.747 × 10⁹⁹(100-digit number)

27473094815003360407…00634711867708602401

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

Level 4 — Twin Prime Pair (2^4 × origin ± 1)

2^4 × origin − 1

5.494 × 10⁹⁹(100-digit number)

54946189630006720815…01269423735417204799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

5.494 × 10⁹⁹(100-digit number)

54946189630006720815…01269423735417204801

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 consecutive prime numbers forming a Bi-Twin Chain. The prime chain origin — the large number shown above — anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

★★☆☆☆

Rarity

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

How Primecoin's Proof-of-Work Constructs These Primes

Primecoin stores a value called the prime chain origin in each block. The miner found a large integer such that when divided by a primorial (the product of small primes: 2 × 3 × 5 × 7 × …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial — that divisibility is part of the proof.

Prime Chain Origin = First Prime × Primorial (2·3·5·7·11·…)

Source: Primecoin prime.cpp — CheckPrimeProofOfWork()

This is why the origin has many small prime factors — those factors are the primorial divisor. The chain then extends from the first prime using the TWN formula:

TWN: twin pairs (p, p+2) where p = origin/primorial − 1 and p+2 = origin/primorial + 1

Cross-reference on Chainz Explorer

Block #416,931

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