Block #135,678

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 8/26/2013, 6:38:49 PM · Difficulty 9.8119 · 6,691,422 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

77cc3b320a156861fc277ca9259c8592ced8e773ddb81c70e56d7e91deb78985

View on Chainz ↗

Height

#135,678

Difficulty

9.811887

Transactions

Size

5.52 KB

Version

Bits

09cfd7db

Nonce

30,151

Timestamp

8/26/2013, 6:38:49 PM

Confirmations

6,691,422

Mined by

⛏️ P2SHAZxbBNyfUXrYABCdaXJ6pzXUZBfG5TfXgf

Merkle Root

1f489065f8c67e98665159da9886a9c40dc95f8a211f34333c641eb2a1d55d17

Transactions (3)

Coinbase45c84eee86d13a…eed20f406805b3

1 in → 1 out10.4400 XPM109 B

AZxbBNyf…fG5TfXgf

17530e6e29e454…14267398f8e3c3

2 in → 1 out4.0064 XPM340 B

AULNAWJJ…uphHNHuT

40d6cc15682268…4d89f86350e7e6

34 in → 2 out78.5910 XPM5.00 KB

AZAMmZ4r…gZdfuiKv AdrZgAo5…aLTu4NiJ

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.

2.437 × 10⁹³(94-digit number)

24379380962114179249…98395632918717120499

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

2.437 × 10⁹³(94-digit number)

24379380962114179249…98395632918717120499

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.437 × 10⁹³(94-digit number)

24379380962114179249…98395632918717120501

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

4.875 × 10⁹³(94-digit number)

48758761924228358499…96791265837434240999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.875 × 10⁹³(94-digit number)

48758761924228358499…96791265837434241001

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

9.751 × 10⁹³(94-digit number)

97517523848456716999…93582531674868481999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.751 × 10⁹³(94-digit number)

97517523848456716999…93582531674868482001

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

1.950 × 10⁹⁴(95-digit number)

19503504769691343399…87165063349736963999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.950 × 10⁹⁴(95-digit number)

19503504769691343399…87165063349736964001

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

3.900 × 10⁹⁴(95-digit number)

39007009539382686799…74330126699473927999

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

Block #135,678

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