Block #135,717

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 8/26/2013, 7:14:34 PM · Difficulty 9.8121 · 6,669,092 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

896cd4d7cb870b2f99049d909cb3cc0fb9a85bcc92dba99f3e5360ea84181c17

View on Chainz ↗

Height

#135,717

Difficulty

9.812059

Transactions

Size

653 B

Version

Bits

09cfe31a

Nonce

34,385

Timestamp

8/26/2013, 7:14:34 PM

Confirmations

6,669,092

Mined by

⛏️ P2SHAJnQwxPihM5Er19MSSTNpiZh7R4ZkgQkNk

Merkle Root

cc4635790d81dd941c2cf5eedd0c204e4b7f0b1203472cb2b95aeb90bb916e95

Transactions (3)

Coinbase9f326217789d4b…f983aa0d49a5ed

1 in → 1 out10.3900 XPM109 B

AJnQwxPi…4ZkgQkNk

9c45c251701ce0…1f72c58d6a9224

1 in → 2 out6.2674 XPM227 B

AdwUeLh2…J1qTNCR4 AG4ofzC9…ieooZRox

f1775135bd1b8a…4a3f323053596a

1 in → 2 out5.3442 XPM226 B

AHjQfvKD…HK5TRF7j AS5xuHTU…4uW2BWq3

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.

4.242 × 10⁹⁶(97-digit number)

42425356642537897735…24950813598139033469

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

4.242 × 10⁹⁶(97-digit number)

42425356642537897735…24950813598139033469

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.242 × 10⁹⁶(97-digit number)

42425356642537897735…24950813598139033471

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

8.485 × 10⁹⁶(97-digit number)

84850713285075795471…49901627196278066939

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.485 × 10⁹⁶(97-digit number)

84850713285075795471…49901627196278066941

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

1.697 × 10⁹⁷(98-digit number)

16970142657015159094…99803254392556133879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.697 × 10⁹⁷(98-digit number)

16970142657015159094…99803254392556133881

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

3.394 × 10⁹⁷(98-digit number)

33940285314030318188…99606508785112267759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.394 × 10⁹⁷(98-digit number)

33940285314030318188…99606508785112267761

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

6.788 × 10⁹⁷(98-digit number)

67880570628060636377…99213017570224535519

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.788 × 10⁹⁷(98-digit number)

67880570628060636377…99213017570224535521

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