Block #218,339

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/19/2013, 9:02:13 PM · Difficulty 9.9303 · 6,573,685 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

fa83cf6dc728f841e12adaa9876d51b9dfdb14815a307856dc212c6c2732f614

View on Chainz ↗

Height

#218,339

Difficulty

9.930336

Transactions

Size

906 B

Version

Bits

09ee2a86

Nonce

23,114

Timestamp

10/19/2013, 9:02:13 PM

Confirmations

6,573,685

Merkle Root

b4a4932402041e6b25d48b0ed9c05fe264e759a8feb8dff14819428caf086e0e

Transactions (2)

Coinbase6c105b57676498…af01057d9480e6

1 in → 1 out10.1400 XPM110 B

AbuU1HhS…JPwsJEbB

f035fefbbc8e75…496c082be55e4c

4 in → 7 out40.6400 XPM706 B

AaezqYHQ…NhR8pjrZ AbkuBBVh…fojLj8yS APd3tden…qhRE2ARG AdToHhys…uNu3dqWo ARTJCDz1…oJ6Hi1d9

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.

4.161 × 10⁹⁵(96-digit number)

41615775389647753357…06122932429200647679

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

4.161 × 10⁹⁵(96-digit number)

41615775389647753357…06122932429200647679

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.161 × 10⁹⁵(96-digit number)

41615775389647753357…06122932429200647681

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

8.323 × 10⁹⁵(96-digit number)

83231550779295506715…12245864858401295359

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.323 × 10⁹⁵(96-digit number)

83231550779295506715…12245864858401295361

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

1.664 × 10⁹⁶(97-digit number)

16646310155859101343…24491729716802590719

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.664 × 10⁹⁶(97-digit number)

16646310155859101343…24491729716802590721

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

3.329 × 10⁹⁶(97-digit number)

33292620311718202686…48983459433605181439

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.329 × 10⁹⁶(97-digit number)

33292620311718202686…48983459433605181441

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

6.658 × 10⁹⁶(97-digit number)

66585240623436405372…97966918867210362879

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