Block #162,712

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 9/13/2013, 12:17:12 PM · Difficulty 9.8613 · 6,642,343 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

39cff4f88d6c9ca943500f8b9c9b973d1015a8f59981325d4f418ba7ac19334c

View on Chainz ↗

Height

#162,712

Difficulty

9.861273

Transactions

Size

798 B

Version

Bits

09dc7c64

Nonce

30,871

Timestamp

9/13/2013, 12:17:12 PM

Confirmations

6,642,343

Mined by

⛏️ P2SHAdDUNTYmz3S4v6JdhjgL2fyWhUv4Zjqa77

Merkle Root

ab8d89dbfc7fc827cad46d975a90115fcba7fe335def66a8efc4676a92e60ea7

Transactions (3)

Coinbase77c1ec23ce765b…905073177aaae1

1 in → 1 out10.2900 XPM109 B

AdDUNTYm…v4Zjqa77

6c485b7a4b4618…85f6bb28f14d2c

2 in → 2 out110.5072 XPM374 B

AKqTVeeg…TNFpeiGJ Aeg3BHTE…Qg3NHgtE

2ffdd07e1ad75d…974466289d0ce5

1 in → 2 out2.0114 XPM226 B

AdJHx7py…Enebbbhd AXRFzZCA…qn7X1N6y

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.

2.410 × 10⁹¹(92-digit number)

24106131456638415378…13408079722246553599

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

2.410 × 10⁹¹(92-digit number)

24106131456638415378…13408079722246553599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.410 × 10⁹¹(92-digit number)

24106131456638415378…13408079722246553601

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

4.821 × 10⁹¹(92-digit number)

48212262913276830756…26816159444493107199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.821 × 10⁹¹(92-digit number)

48212262913276830756…26816159444493107201

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

9.642 × 10⁹¹(92-digit number)

96424525826553661513…53632318888986214399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.642 × 10⁹¹(92-digit number)

96424525826553661513…53632318888986214401

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

1.928 × 10⁹²(93-digit number)

19284905165310732302…07264637777972428799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.928 × 10⁹²(93-digit number)

19284905165310732302…07264637777972428801

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

3.856 × 10⁹²(93-digit number)

38569810330621464605…14529275555944857599

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.856 × 10⁹²(93-digit number)

38569810330621464605…14529275555944857601

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