Block #851,972

TWNLength 11★★★☆☆

Bi-Twin Chain · Discovered 12/13/2014, 5:26:06 PM · Difficulty 10.9702 · 5,986,866 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

75ea33fa9f0ed5629cb427603ca54c3fd25514c624deb3ce349745d4d74e2e2d

View on Chainz ↗

Height

#851,972

Difficulty

10.970239

Transactions

Size

433 B

Version

Bits

0af86199

Nonce

669,037,595

Timestamp

12/13/2014, 5:26:06 PM

Confirmations

5,986,866

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

4ecb5faff81cfc4bc96f13922651003e0570ae03c19213005cabae1ff539cdd3

Transactions (2)

Coinbaseccadaba3ce711e…d1b0eb1dda59d9

1 in → 1 out8.3150 XPM116 B

ANSXnLY2…Sd25TjkD

b7d614110837b3…92f85c9c84883f

1 in → 2 out0.0618 XPM225 B

AHpohnoV…HDp4K8aB ANo8jKhF…Pm6nJEpb

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.

5.708 × 10⁹⁹(100-digit number)

57088789467832563616…74721522672729456639

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

5.708 × 10⁹⁹(100-digit number)

57088789467832563616…74721522672729456639

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.708 × 10⁹⁹(100-digit number)

57088789467832563616…74721522672729456641

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

1.141 × 10¹⁰⁰(101-digit number)

11417757893566512723…49443045345458913279

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.141 × 10¹⁰⁰(101-digit number)

11417757893566512723…49443045345458913281

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

2.283 × 10¹⁰⁰(101-digit number)

22835515787133025446…98886090690917826559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.283 × 10¹⁰⁰(101-digit number)

22835515787133025446…98886090690917826561

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

4.567 × 10¹⁰⁰(101-digit number)

45671031574266050893…97772181381835653119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.567 × 10¹⁰⁰(101-digit number)

45671031574266050893…97772181381835653121

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

9.134 × 10¹⁰⁰(101-digit number)

91342063148532101786…95544362763671306239

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

9.134 × 10¹⁰⁰(101-digit number)

91342063148532101786…95544362763671306241

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

Level 5 — Twin Prime Pair (2^5 × origin ± 1)

2^5 × origin − 1

1.826 × 10¹⁰¹(102-digit number)

18268412629706420357…91088725527342612479

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 11 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

RareChain length 11

Approximately 1 in 1,000 blocks. Noteworthy discoveries.

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 #851,972

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