Block #138,224

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 8/28/2013, 7:17:23 AM · Difficulty 9.8246 · 6,688,426 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f1f78c8b7740695ce0f7a3b4eabeb8d3704888295d3c333242df4803f8598d66

View on Chainz ↗

Height

#138,224

Difficulty

9.824579

Transactions

Size

1.36 KB

Version

Bits

09d31796

Nonce

55,617

Timestamp

8/28/2013, 7:17:23 AM

Confirmations

6,688,426

Mined by

⛏️ P2SHAdBiCnrF5tcJSk2P8X7TAzXHP4WAmAgtoo

Merkle Root

fa9dab0492e8ba5d139ba699d595337ae7dc7eaee2b422c8199719ccc085d31b

Transactions (3)

Coinbasedfc922886c9d88…1fd4ecef2aacab

1 in → 1 out10.3600 XPM109 B

AdBiCnrF…WAmAgtoo

cf8d667ba37d27…8e1d92dc8089a0

5 in → 2 out349.9100 XPM818 B

AauD3ke7…djJKjxdK ALZodJTs…xfSqGtat

a9686c2ef4d4a0…5c59a3097e7bd7

2 in → 2 out12.0578 XPM375 B

AaamjEDB…UqchbkWj AQMAuDYp…RcS2bnY6

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.

1.211 × 10⁹⁰(91-digit number)

12111203663005884047…66768304404177252639

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

1.211 × 10⁹⁰(91-digit number)

12111203663005884047…66768304404177252639

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.211 × 10⁹⁰(91-digit number)

12111203663005884047…66768304404177252641

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

2.422 × 10⁹⁰(91-digit number)

24222407326011768094…33536608808354505279

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.422 × 10⁹⁰(91-digit number)

24222407326011768094…33536608808354505281

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

4.844 × 10⁹⁰(91-digit number)

48444814652023536188…67073217616709010559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.844 × 10⁹⁰(91-digit number)

48444814652023536188…67073217616709010561

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

9.688 × 10⁹⁰(91-digit number)

96889629304047072376…34146435233418021119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

9.688 × 10⁹⁰(91-digit number)

96889629304047072376…34146435233418021121

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

1.937 × 10⁹¹(92-digit number)

19377925860809414475…68292870466836042239

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 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

CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

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 #138,224

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