Block #138,662

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 8/28/2013, 1:11:43 PM · Difficulty 9.8274 · 6,686,954 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

a39fad150286924f7c4f946464d427551ccd860d771bb27e0e94866ad71e2985

View on Chainz ↗

Height

#138,662

Difficulty

9.827445

Transactions

Size

919 B

Version

Bits

09d3d36d

Nonce

16,492

Timestamp

8/28/2013, 1:11:43 PM

Confirmations

6,686,954

Mined by

⛏️ P2SHAXYmrbdyE35zVW32TW2Pso8Ewjf3CJJzwR

Merkle Root

1ef779fb91c2322c2b4745403d1575ca8da7c1d298275ad64772ca66d2ede945

Transactions (4)

Coinbase105b5988b397cd…714a8d85afe69d

1 in → 1 out10.3700 XPM109 B

AXYmrbdy…f3CJJzwR

360a662d8a4f95…f8bfa802cae9ed

2 in → 2 out20.8700 XPM305 B

AVWB3Ybr…i5Z8mGNq AMEZCfKf…y1D9vn7E

9de066ac3d3f0f…b93fdf93fce5a7

1 in → 2 out10.3700 XPM191 B

AXiuhqLM…BnWxzTFY AFz9fXym…wh4UqpkL

0eeb51e0755d16…ad377141062dfd

1 in → 2 out1.6083 XPM225 B

ANtrU6Kh…zi49XtJw AcBKhhYn…sdFuuArV

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.

3.815 × 10⁹¹(92-digit number)

38152870354081955522…08813251596632392249

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

3.815 × 10⁹¹(92-digit number)

38152870354081955522…08813251596632392249

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.815 × 10⁹¹(92-digit number)

38152870354081955522…08813251596632392251

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

7.630 × 10⁹¹(92-digit number)

76305740708163911044…17626503193264784499

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

7.630 × 10⁹¹(92-digit number)

76305740708163911044…17626503193264784501

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

1.526 × 10⁹²(93-digit number)

15261148141632782208…35253006386529568999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.526 × 10⁹²(93-digit number)

15261148141632782208…35253006386529569001

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

3.052 × 10⁹²(93-digit number)

30522296283265564417…70506012773059137999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.052 × 10⁹²(93-digit number)

30522296283265564417…70506012773059138001

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

6.104 × 10⁹²(93-digit number)

61044592566531128835…41012025546118275999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.104 × 10⁹²(93-digit number)

61044592566531128835…41012025546118276001

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 #138,662

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