Block #654,103

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.

5.815 × 10⁹⁷(98-digit number)

58155943873598571952…99801833707541804799

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

5.815 × 10⁹⁷(98-digit number)

58155943873598571952…99801833707541804799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.815 × 10⁹⁷(98-digit number)

58155943873598571952…99801833707541804801

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

1.163 × 10⁹⁸(99-digit number)

11631188774719714390…99603667415083609599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.163 × 10⁹⁸(99-digit number)

11631188774719714390…99603667415083609601

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

2.326 × 10⁹⁸(99-digit number)

23262377549439428780…99207334830167219199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.326 × 10⁹⁸(99-digit number)

23262377549439428780…99207334830167219201

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

4.652 × 10⁹⁸(99-digit number)

46524755098878857561…98414669660334438399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.652 × 10⁹⁸(99-digit number)

46524755098878857561…98414669660334438401

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

9.304 × 10⁹⁸(99-digit number)

93049510197757715123…96829339320668876799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

9.304 × 10⁹⁸(99-digit number)

93049510197757715123…96829339320668876801

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

Level 5 — Twin Prime Pair (2^5 × origin ± 1)

2^5 × origin − 1

1.860 × 10⁹⁹(100-digit number)

18609902039551543024…93658678641337753599

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 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

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 TWN formula:

TWN: twin pairs (p, p+2) where p = origin/primorial − 1 and p+2 = origin/primorial + 1

Cross-reference on Chainz Explorer