Block #3,082,731

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.

6.459 × 10⁹⁴(95-digit number)

64598362347873716000…43122994110609066701

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

6.459 × 10⁹⁴(95-digit number)

64598362347873716000…43122994110609066701

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^1 × origin + 1

1.291 × 10⁹⁵(96-digit number)

12919672469574743200…86245988221218133401

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^2 × origin + 1

2.583 × 10⁹⁵(96-digit number)

25839344939149486400…72491976442436266801

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^3 × origin + 1

5.167 × 10⁹⁵(96-digit number)

51678689878298972800…44983952884872533601

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^4 × origin + 1

1.033 × 10⁹⁶(97-digit number)

10335737975659794560…89967905769745067201

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^5 × origin + 1

2.067 × 10⁹⁶(97-digit number)

20671475951319589120…79935811539490134401

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^6 × origin + 1

4.134 × 10⁹⁶(97-digit number)

41342951902639178240…59871623078980268801

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^7 × origin + 1

8.268 × 10⁹⁶(97-digit number)

82685903805278356480…19743246157960537601

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^8 × origin + 1

1.653 × 10⁹⁷(98-digit number)

16537180761055671296…39486492315921075201

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^9 × origin + 1

3.307 × 10⁹⁷(98-digit number)

33074361522111342592…78972984631842150401

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^10 × origin + 1

6.614 × 10⁹⁷(98-digit number)

66148723044222685184…57945969263684300801

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 11 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

RareChain length 11

Approximately 1 in 1,000 blocks. Noteworthy discoveries.

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 #3,082,731

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