Block #309,342

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/13/2013, 1:37:56 PM · Difficulty 9.9948 · 6,485,746 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

afb30d5da6f00c1f0b268e287491de3e98373991bb7016a32232f00fc98afa06

View on Chainz ↗

Height

#309,342

Difficulty

9.994791

Transactions

Size

1.66 KB

Version

Bits

09feaaa1

Nonce

12,760

Timestamp

12/13/2013, 1:37:56 PM

Confirmations

6,485,746

Mined by

⛏️ ^aRAJzWx2VDShfKP9pKK3jZqTbn4sj4ZSMit5

Merkle Root

78880aa5b24c1be975e2840f6103b9bf5ef2a0a1f3cee46c15821f183b0423ff

Transactions (2)

Coinbasec56ce2cd077908…e43f4798f041ea

1 in → 30 out9.9800 XPM1.06 KB

AJzWx2VD…j4ZSMit5 AcM3ext6…Hwtj9Qp3 ASWX42VM…XypQABkY AQjgN2dS…uEQws1Mu AebWhrRm…K61Qrkws

1a64e6bd15ad80…d0db97152ab689

3 in → 2 out2.9941 XPM521 B

ANEy3CVX…x7CkqKeM ANxC5bHf…YByrhgt6

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.

4.416 × 10⁹⁷(98-digit number)

44164273691937429157…88269324459384792849

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

4.416 × 10⁹⁷(98-digit number)

44164273691937429157…88269324459384792849

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.416 × 10⁹⁷(98-digit number)

44164273691937429157…88269324459384792851

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

8.832 × 10⁹⁷(98-digit number)

88328547383874858314…76538648918769585699

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.832 × 10⁹⁷(98-digit number)

88328547383874858314…76538648918769585701

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

1.766 × 10⁹⁸(99-digit number)

17665709476774971662…53077297837539171399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.766 × 10⁹⁸(99-digit number)

17665709476774971662…53077297837539171401

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

3.533 × 10⁹⁸(99-digit number)

35331418953549943325…06154595675078342799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.533 × 10⁹⁸(99-digit number)

35331418953549943325…06154595675078342801

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

7.066 × 10⁹⁸(99-digit number)

70662837907099886651…12309191350156685599

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

7.066 × 10⁹⁸(99-digit number)

70662837907099886651…12309191350156685601

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