Block #413,014

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.

4.009 × 10¹⁰¹(102-digit number)

40098325482360718029…68998714278578717441

Discovered Prime Numbers

p_k = 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.

origin + 1

4.009 × 10¹⁰¹(102-digit number)

40098325482360718029…68998714278578717441

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^1 × origin + 1

8.019 × 10¹⁰¹(102-digit number)

80196650964721436059…37997428557157434881

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^2 × origin + 1

1.603 × 10¹⁰²(103-digit number)

16039330192944287211…75994857114314869761

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^3 × origin + 1

3.207 × 10¹⁰²(103-digit number)

32078660385888574423…51989714228629739521

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^4 × origin + 1

6.415 × 10¹⁰²(103-digit number)

64157320771777148847…03979428457259479041

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^5 × origin + 1

1.283 × 10¹⁰³(104-digit number)

12831464154355429769…07958856914518958081

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^6 × origin + 1

2.566 × 10¹⁰³(104-digit number)

25662928308710859539…15917713829037916161

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^7 × origin + 1

5.132 × 10¹⁰³(104-digit number)

51325856617421719078…31835427658075832321

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^8 × origin + 1

1.026 × 10¹⁰⁴(105-digit number)

10265171323484343815…63670855316151664641

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^9 × origin + 1

2.053 × 10¹⁰⁴(105-digit number)

20530342646968687631…27341710632303329281

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 10 consecutive prime numbers forming a Cunningham Chain of the Second Kind. 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 2CC formula:

2CC: p₁ (first prime), p₂ = 2p₁ − 1, p₃ = 2p₂ − 1, …

Cross-reference on Chainz Explorer

Block #413,014

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