Block #316,654

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/17/2013, 5:07:29 AM · Difficulty 10.1392 · 6,489,406 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

c4b9ac3f3b7b54ec4f74716dd3fa0aaf0ac582c38595f69dd760905d86d452f4

View on Chainz ↗

Height

#316,654

Difficulty

10.139150

Transactions

Size

1.73 KB

Version

Bits

0a239f58

Nonce

105,369

Timestamp

12/17/2013, 5:07:29 AM

Confirmations

6,489,406

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

efde3b9ab91c4ae57486e55957b02ebac3d6d8f3d9e413204a8d42f0d3f21caa

Transactions (3)

Coinbasedc07750df2b044…4173f675752f5a

1 in → 1 out9.7412 XPM110 B

AeKNrjNj…1NEq95Ta

b97974ea1d9767…6c86a74fdb3dee

2 in → 1 out35.0600 XPM339 B

AUZssAvZ…AEbuKnVg

6a4a9e8e8fd9ca…379600af753401

8 in → 1 out2.7300 XPM1.20 KB

AVQEFSrn…1EK8ojym

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.

3.740 × 10¹⁰⁵(106-digit number)

37407465478758977632…11807640889230633599

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

3.740 × 10¹⁰⁵(106-digit number)

37407465478758977632…11807640889230633599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.740 × 10¹⁰⁵(106-digit number)

37407465478758977632…11807640889230633601

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

7.481 × 10¹⁰⁵(106-digit number)

74814930957517955264…23615281778461267199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

7.481 × 10¹⁰⁵(106-digit number)

74814930957517955264…23615281778461267201

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

1.496 × 10¹⁰⁶(107-digit number)

14962986191503591052…47230563556922534399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.496 × 10¹⁰⁶(107-digit number)

14962986191503591052…47230563556922534401

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

2.992 × 10¹⁰⁶(107-digit number)

29925972383007182105…94461127113845068799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.992 × 10¹⁰⁶(107-digit number)

29925972383007182105…94461127113845068801

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

5.985 × 10¹⁰⁶(107-digit number)

59851944766014364211…88922254227690137599

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

5.985 × 10¹⁰⁶(107-digit number)

59851944766014364211…88922254227690137601

Verify on FactorDB ↗Wolfram Alpha ↗

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

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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