Block #414,441

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.

1.542 × 10⁹⁹(100-digit number)

15421120855198614422…37489199325275494399

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

1.542 × 10⁹⁹(100-digit number)

15421120855198614422…37489199325275494399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.542 × 10⁹⁹(100-digit number)

15421120855198614422…37489199325275494401

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

3.084 × 10⁹⁹(100-digit number)

30842241710397228845…74978398650550988799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.084 × 10⁹⁹(100-digit number)

30842241710397228845…74978398650550988801

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

6.168 × 10⁹⁹(100-digit number)

61684483420794457690…49956797301101977599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.168 × 10⁹⁹(100-digit number)

61684483420794457690…49956797301101977601

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

1.233 × 10¹⁰⁰(101-digit number)

12336896684158891538…99913594602203955199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.233 × 10¹⁰⁰(101-digit number)

12336896684158891538…99913594602203955201

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

2.467 × 10¹⁰⁰(101-digit number)

24673793368317783076…99827189204407910399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.467 × 10¹⁰⁰(101-digit number)

24673793368317783076…99827189204407910401

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 #414,441

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