Block #311,088

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 9:39:10 AM · Difficulty 9.9954 · 6,493,108 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

2a400346130e7da18f63f8d0c06f50bd96e394836c99de7f5679ccfc2eb52953

View on Chainz ↗

Height

#311,088

Difficulty

9.995374

Transactions

Size

1.15 KB

Version

Bits

09fed0d5

Nonce

121,420

Timestamp

12/14/2013, 9:39:10 AM

Confirmations

6,493,108

Mined by

⛏️ 0aR':RAc6mGvmQ9ya8dQxqWNQYGz81U4XqWmPHjR

Merkle Root

f401e86dfab4c0b5f2dc60e9111656cc6d37c2c49cd8e697b8c08d51a6e705e4

Transactions (1)

Coinbasef401e86dfab4c0…c08d51a6e705e4

1 in → 30 out9.9700 XPM1.06 KB

Ac6mGvmQ…XqWmPHjR ATiP2myP…qVUH8Grb ANyaNSVA…ybz5ygRU AcCB46P6…9J8bgWwi AHjxjypg…BfqPH2LH

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.

4.774 × 10⁹⁷(98-digit number)

47745093208536565362…18654232506412476479

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

4.774 × 10⁹⁷(98-digit number)

47745093208536565362…18654232506412476479

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.774 × 10⁹⁷(98-digit number)

47745093208536565362…18654232506412476481

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

9.549 × 10⁹⁷(98-digit number)

95490186417073130724…37308465012824952959

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

9.549 × 10⁹⁷(98-digit number)

95490186417073130724…37308465012824952961

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

1.909 × 10⁹⁸(99-digit number)

19098037283414626144…74616930025649905919

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.909 × 10⁹⁸(99-digit number)

19098037283414626144…74616930025649905921

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

3.819 × 10⁹⁸(99-digit number)

38196074566829252289…49233860051299811839

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.819 × 10⁹⁸(99-digit number)

38196074566829252289…49233860051299811841

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

7.639 × 10⁹⁸(99-digit number)

76392149133658504579…98467720102599623679

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