Block #691,293

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 8/24/2014, 10:06:41 PM · Difficulty 10.9552 · 6,119,706 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

54f0245f2e0f36a337362f91caeb35381602696a3b03d5ed2cee4f2823a205da

View on Chainz ↗

Height

#691,293

Difficulty

10.955173

Transactions

Size

501 B

Version

Bits

0af48638

Nonce

1,595,871,361

Timestamp

8/24/2014, 10:06:41 PM

Confirmations

6,119,706

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

28b295842e1256ad6c9cd79ff64b456df50ddaf28bf6a7fa3c59d1a45b747815

Transactions (2)

Coinbase16dd61e23ea980…46b2a2fc4be91e

1 in → 1 out8.3300 XPM116 B

ANSXnLY2…Sd25TjkD

026a8e6c86b898…80eac5082eb8b9

1 in → 5 out8.3100 XPM294 B

AcBvLaSu…X3gaHvJ1 ASc5A66o…rgEAg76F AFtKUKUa…WyLEn4xB AYk6SdQf…UGcYbb4t AeKNrjNj…1NEq95Ta

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.

3.306 × 10⁹⁶(97-digit number)

33066722254449217838…41308686807230440959

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

3.306 × 10⁹⁶(97-digit number)

33066722254449217838…41308686807230440959

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.306 × 10⁹⁶(97-digit number)

33066722254449217838…41308686807230440961

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

6.613 × 10⁹⁶(97-digit number)

66133444508898435677…82617373614460881919

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.613 × 10⁹⁶(97-digit number)

66133444508898435677…82617373614460881921

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

1.322 × 10⁹⁷(98-digit number)

13226688901779687135…65234747228921763839

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.322 × 10⁹⁷(98-digit number)

13226688901779687135…65234747228921763841

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

2.645 × 10⁹⁷(98-digit number)

26453377803559374271…30469494457843527679

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.645 × 10⁹⁷(98-digit number)

26453377803559374271…30469494457843527681

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

5.290 × 10⁹⁷(98-digit number)

52906755607118748542…60938988915687055359

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

5.290 × 10⁹⁷(98-digit number)

52906755607118748542…60938988915687055361

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 #691,293

Cookie Preferences