Block #165,012

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.

1.938 × 10⁹⁴(95-digit number)

19389344897940487082…93304784393055031681

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

1.938 × 10⁹⁴(95-digit number)

19389344897940487082…93304784393055031681

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^1 × origin + 1

3.877 × 10⁹⁴(95-digit number)

38778689795880974164…86609568786110063361

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^2 × origin + 1

7.755 × 10⁹⁴(95-digit number)

77557379591761948329…73219137572220126721

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^3 × origin + 1

1.551 × 10⁹⁵(96-digit number)

15511475918352389665…46438275144440253441

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^4 × origin + 1

3.102 × 10⁹⁵(96-digit number)

31022951836704779331…92876550288880506881

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^5 × origin + 1

6.204 × 10⁹⁵(96-digit number)

62045903673409558663…85753100577761013761

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^6 × origin + 1

1.240 × 10⁹⁶(97-digit number)

12409180734681911732…71506201155522027521

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^7 × origin + 1

2.481 × 10⁹⁶(97-digit number)

24818361469363823465…43012402311044055041

Verify on FactorDB ↗Wolfram Alpha ↗

×2−1 →

2^8 × origin + 1

4.963 × 10⁹⁶(97-digit number)

49636722938727646930…86024804622088110081

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 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

CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

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