Block #657,607

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 8/1/2014, 8:25:12 AM · Difficulty 10.9562 · 6,156,707 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

a98c6d619fb9e316d4e7085938294122930e768a96d0be2fc3d31ede045698f9

View on Chainz ↗

Height

#657,607

Difficulty

10.956214

Transactions

Size

17.55 KB

Version

Bits

0af4ca76

Nonce

1,830,860,134

Timestamp

8/1/2014, 8:25:12 AM

Confirmations

6,156,707

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

97ad469a1c1ba47cefa6fbbc683162097ba6c66fc5ec300e1fa2c63b9ae01844

Transactions (3)

Coinbase84c4622ecaa652…9ba5ec537e6180

1 in → 1 out8.5100 XPM116 B

ANSXnLY2…Sd25TjkD

c07f10a2003493…7a77c85b1c755f

2 in → 2 out11.6473 XPM374 B

AVEwmYd1…5VePBcAF AMv83Zpd…VArcgfdL

5ec7017f1c07d8…be7551559d6e57

117 in → 2 out62.0101 XPM16.99 KB

AbLPMsjH…AYWazRqs APP97DSB…Wfj7mmNu

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.

7.442 × 10⁹⁷(98-digit number)

74427692091765214596…51319392873671147519

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

7.442 × 10⁹⁷(98-digit number)

74427692091765214596…51319392873671147519

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.442 × 10⁹⁷(98-digit number)

74427692091765214596…51319392873671147521

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

1.488 × 10⁹⁸(99-digit number)

14885538418353042919…02638785747342295039

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.488 × 10⁹⁸(99-digit number)

14885538418353042919…02638785747342295041

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

2.977 × 10⁹⁸(99-digit number)

29771076836706085838…05277571494684590079

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.977 × 10⁹⁸(99-digit number)

29771076836706085838…05277571494684590081

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

5.954 × 10⁹⁸(99-digit number)

59542153673412171677…10555142989369180159

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.954 × 10⁹⁸(99-digit number)

59542153673412171677…10555142989369180161

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

1.190 × 10⁹⁹(100-digit number)

11908430734682434335…21110285978738360319

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.190 × 10⁹⁹(100-digit number)

11908430734682434335…21110285978738360321

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

Block #657,607

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