Block #2,640,513

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.

2.088 × 10⁹⁷(98-digit number)

20881912855744163069…50955522154826239999

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

2.088 × 10⁹⁷(98-digit number)

20881912855744163069…50955522154826239999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^1 × origin − 1

4.176 × 10⁹⁷(98-digit number)

41763825711488326138…01911044309652479999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^2 × origin − 1

8.352 × 10⁹⁷(98-digit number)

83527651422976652277…03822088619304959999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^3 × origin − 1

1.670 × 10⁹⁸(99-digit number)

16705530284595330455…07644177238609919999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^4 × origin − 1

3.341 × 10⁹⁸(99-digit number)

33411060569190660910…15288354477219839999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^5 × origin − 1

6.682 × 10⁹⁸(99-digit number)

66822121138381321821…30576708954439679999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^6 × origin − 1

1.336 × 10⁹⁹(100-digit number)

13364424227676264364…61153417908879359999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^7 × origin − 1

2.672 × 10⁹⁹(100-digit number)

26728848455352528728…22306835817758719999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^8 × origin − 1

5.345 × 10⁹⁹(100-digit number)

53457696910705057457…44613671635517439999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^9 × origin − 1

1.069 × 10¹⁰⁰(101-digit number)

10691539382141011491…89227343271034879999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^10 × origin − 1

2.138 × 10¹⁰⁰(101-digit number)

21383078764282022982…78454686542069759999

Verify on FactorDB ↗Wolfram Alpha ↗

×2+1 →

2^11 × origin − 1

4.276 × 10¹⁰⁰(101-digit number)

42766157528564045965…56909373084139519999

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 12 consecutive prime numbers forming a Cunningham Chain of the First 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

ExceptionalChain length 12

Around 1 in 10,000 blocks. A significant mathematical achievement.

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 1CC formula:

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

Cross-reference on Chainz Explorer

Block #2,640,513

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