Block #135,223

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 8/26/2013, 12:20:55 PM · Difficulty 9.8090 · 6,671,935 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

df5787ac9369f5a54e8ea9f3b3e00d875df923d8d863bc9f2bd61b5ce07d00bc

View on Chainz ↗

Height

#135,223

Difficulty

9.809025

Transactions

Size

1.14 KB

Version

Bits

09cf1c44

Nonce

47,240

Timestamp

8/26/2013, 12:20:55 PM

Confirmations

6,671,935

Mined by

⛏️ P2SHAZxbBNyfUXrYABCdaXJ6pzXUZBfG5TfXgf

Merkle Root

42f43887d43f5ea02314fa9badd8fa1bead85d4ac4d2bff6db9ef5938e7ab41f

Transactions (4)

Coinbase7da6f9798282bf…909fc42437c5ac

1 in → 1 out10.4100 XPM109 B

AZxbBNyf…fG5TfXgf

dcfabb4e598da6…7f525c6e961fdf

2 in → 2 out10.6505 XPM374 B

ARykYKJL…bzF3K4J6 AUdE3Hat…MynJ79VF

88c9bb751686da…676fe8b4348875

1 in → 2 out4.3777 XPM225 B

ANN8gjLH…8ToruqzW AHZntHqt…jVGXQsZE

5506db6ff1c83a…ca1edb102639b0

2 in → 2 out2.7948 XPM374 B

AV1MQrfi…Gjh354UE ATE5eGSA…KVRRcjaW

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.

3.454 × 10⁹³(94-digit number)

34543149263491764787…55276358503972744959

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

3.454 × 10⁹³(94-digit number)

34543149263491764787…55276358503972744959

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.454 × 10⁹³(94-digit number)

34543149263491764787…55276358503972744961

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

6.908 × 10⁹³(94-digit number)

69086298526983529574…10552717007945489919

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.908 × 10⁹³(94-digit number)

69086298526983529574…10552717007945489921

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

1.381 × 10⁹⁴(95-digit number)

13817259705396705914…21105434015890979839

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.381 × 10⁹⁴(95-digit number)

13817259705396705914…21105434015890979841

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

2.763 × 10⁹⁴(95-digit number)

27634519410793411829…42210868031781959679

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.763 × 10⁹⁴(95-digit number)

27634519410793411829…42210868031781959681

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

5.526 × 10⁹⁴(95-digit number)

55269038821586823659…84421736063563919359

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

5.526 × 10⁹⁴(95-digit number)

55269038821586823659…84421736063563919361

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 #135,223

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