Block #291,722

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/3/2013, 8:15:54 AM · Difficulty 9.9900 · 6,511,605 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f0a463418929b466eb699e31afc422d3e6830f02719039f452fbfb0a7af67133

View on Chainz ↗

Height

#291,722

Difficulty

9.989995

Transactions

Size

3.08 KB

Version

Bits

09fd7053

Nonce

13,568

Timestamp

12/3/2013, 8:15:54 AM

Confirmations

6,511,605

Mined by

⛏️ P2SHANgpXsRH3QpBW31v8if88xCudo8C57hWmM

Merkle Root

e7b4dcb1381edf6f4afd714a9e746e7ed7e52aa0bde44d1f74a8e37586778fc1

Transactions (5)

Coinbase286353f18e56f9…79b415cba79015

1 in → 1 out10.0700 XPM109 B

ANgpXsRH…8C57hWmM

c975eaf5221941…e1709674fe6bba

1 in → 2 out10.0900 XPM192 B

AaLBYWdH…e9nxsSZs AJDWc8DM…LhD5nfaV

254036688e49b4…8fb1cb8edfe7c3

2 in → 1 out6.1847 XPM340 B

AMBC6JGb…tZMXhMBC

2dddcddb0a50b2…cfe71a78e674dc

6 in → 18 out60.3390 XPM1.28 KB

AXAHbAo8…Ev6LjF3i AehmFh5z…MJPYDKkx AYYqwMKM…WWSjysXU AUi56htg…pf8nicRC ALX3iX2q…1oQ8ea28

e13b39a8e2ae73…8f480fb0993660

7 in → 2 out21.2167 XPM1.09 KB

AYhYr8Dy…64hPugZR AXZiweVh…mH416iQx

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.

1.908 × 10⁹⁴(95-digit number)

19085719168894710348…94575017612459091199

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

1.908 × 10⁹⁴(95-digit number)

19085719168894710348…94575017612459091199

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.908 × 10⁹⁴(95-digit number)

19085719168894710348…94575017612459091201

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

3.817 × 10⁹⁴(95-digit number)

38171438337789420696…89150035224918182399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.817 × 10⁹⁴(95-digit number)

38171438337789420696…89150035224918182401

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

7.634 × 10⁹⁴(95-digit number)

76342876675578841393…78300070449836364799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

7.634 × 10⁹⁴(95-digit number)

76342876675578841393…78300070449836364801

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

1.526 × 10⁹⁵(96-digit number)

15268575335115768278…56600140899672729599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.526 × 10⁹⁵(96-digit number)

15268575335115768278…56600140899672729601

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

3.053 × 10⁹⁵(96-digit number)

30537150670231536557…13200281799345459199

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.053 × 10⁹⁵(96-digit number)

30537150670231536557…13200281799345459201

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