Block #319,726

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/19/2013, 5:09:25 AM · Difficulty 10.1718 · 6,495,285 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

80541841608936e401a410c056d24e728eb9bd068157444120de60903360fe4e

View on Chainz ↗

Height

#319,726

Difficulty

10.171759

Transactions

Size

1.08 KB

Version

Bits

0a2bf86b

Nonce

4,063

Timestamp

12/19/2013, 5:09:25 AM

Confirmations

6,495,285

Mined by

⛏️ aRAac9Hjdgnw4T4L2csqLecJsmZjP4ktypc3

Merkle Root

eab683e8e24c455fb880f1d863aacb3c1d84c94308aa8b2504b7ab341f6482db

Transactions (1)

Coinbaseeab683e8e24c45…b7ab341f6482db

1 in → 28 out9.6300 XPM1015 B

Aac9Hjdg…P4ktypc3 Ad9pSzHt…cpJ3y2zz AZemWJAo…6jhDtieG Actx8HRL…1LEiodun AG5YAjec…VhA4Aeh1

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.

1.336 × 10⁹⁴(95-digit number)

13365524589383174823…11836219335521408119

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

1.336 × 10⁹⁴(95-digit number)

13365524589383174823…11836219335521408119

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.336 × 10⁹⁴(95-digit number)

13365524589383174823…11836219335521408121

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

2.673 × 10⁹⁴(95-digit number)

26731049178766349647…23672438671042816239

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.673 × 10⁹⁴(95-digit number)

26731049178766349647…23672438671042816241

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

5.346 × 10⁹⁴(95-digit number)

53462098357532699295…47344877342085632479

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.346 × 10⁹⁴(95-digit number)

53462098357532699295…47344877342085632481

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

1.069 × 10⁹⁵(96-digit number)

10692419671506539859…94689754684171264959

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.069 × 10⁹⁵(96-digit number)

10692419671506539859…94689754684171264961

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

2.138 × 10⁹⁵(96-digit number)

21384839343013079718…89379509368342529919

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.138 × 10⁹⁵(96-digit number)

21384839343013079718…89379509368342529921

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 #319,726

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