Block #166,839

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/16/2013, 4:32:38 AM · Difficulty 9.8688 · 6,657,698 confirmations

TWN

Bi-Twin Chain

Interleaved pairs of primes that differ by 2, forming twin prime pairs at each level.

Block Header

Block Hash

b0b0df08e81fe5cff63cf854482931f0e688c022470e297b017540e081739911

View on Chainz ↗

Height

#166,839

Difficulty

9.868751

Transactions

Size

516 B

Version

Bits

09de6677

Nonce

98,862

Timestamp

9/16/2013, 4:32:38 AM

Confirmations

6,657,698

Mined by

⛏️ P2SHATcPmT1ArCmhkFFVgkw6zb55EGPwdkEgSY

Merkle Root

e304a0f4a87debf8ce16fc49f4eb1cb0f6186e4229c17fb71413bb518491d00d

Transactions (3)

Coinbase2a558853e68eae…eca10931b5d527

1 in → 1 out10.2700 XPM109 B

ATcPmT1A…PwdkEgSY

55e59a5ce78fb7…634ee1f6cac79b

1 in → 1 out10.2600 XPM158 B

AKN6pNCw…MeATKM2o

f7e4770f9984a2…768fb0cd94fb14

1 in → 1 out10.2700 XPM159 B

Aa571JXD…1d4PwSHX

Prime Chain Origin

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.

3.737 × 10⁹³(94-digit number)

37373850454220459329…56548061227417433599

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

3.737 × 10⁹³(94-digit number)

37373850454220459329…56548061227417433599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.737 × 10⁹³(94-digit number)

37373850454220459329…56548061227417433601

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

7.474 × 10⁹³(94-digit number)

74747700908440918658…13096122454834867199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

7.474 × 10⁹³(94-digit number)

74747700908440918658…13096122454834867201

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

1.494 × 10⁹⁴(95-digit number)

14949540181688183731…26192244909669734399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.494 × 10⁹⁴(95-digit number)

14949540181688183731…26192244909669734401

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

2.989 × 10⁹⁴(95-digit number)

29899080363376367463…52384489819339468799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.989 × 10⁹⁴(95-digit number)

29899080363376367463…52384489819339468801

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

5.979 × 10⁹⁴(95-digit number)

59798160726752734926…04768979638678937599

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 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

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 TWN formula:

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

Cross-reference on Chainz Explorer

Block #166,839

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