Block #251,961

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/9/2013, 7:46:05 AM · Difficulty 9.9705 · 6,557,770 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

998636f28d37dd743cf6905e5568a182ed269fb2097be08e8e13b767f4059302

View on Chainz ↗

Height

#251,961

Difficulty

9.970536

Transactions

Size

1.17 KB

Version

Bits

09f8750a

Nonce

3,041

Timestamp

11/9/2013, 7:46:05 AM

Confirmations

6,557,770

Mined by

⛏️ P2SHAZDUEbdSvp8cw8AAZ1fERZUqSYRYKMRZET

Merkle Root

206fe8ac5876a4e01ddab5060fd305b0247c3c7cf144929c210d25f498c5c32d

Transactions (4)

Coinbase8b55fb505bb58c…8545fbeb8dd14c

1 in → 2 out10.0700 XPM136 B

AZDUEbdS…RYKMRZET AU6kq4CL…dQBLvxLp

b0add5135a587f…cdb2bbf25a9b3c

3 in → 2 out270.0101 XPM523 B

APhZbt91…ERWP9zpq AQauHrwu…NhVfmRxB

7b63cc8bc4086c…b1bc3bf1bddede

1 in → 2 out3.0211 XPM225 B

ARskhKeb…UgDvvX9P Ae6Cn5Jz…eaHJ2PkH

dbb32f21ff7d1a…31fd39c05c47ac

1 in → 2 out5.0142 XPM226 B

AQqre4AZ…4wffyzjk AbmD9gBP…qL4DkdRH

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.

1.080 × 10⁹⁴(95-digit number)

10803187435622716522…21834047368088538699

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

1.080 × 10⁹⁴(95-digit number)

10803187435622716522…21834047368088538699

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.080 × 10⁹⁴(95-digit number)

10803187435622716522…21834047368088538701

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

2.160 × 10⁹⁴(95-digit number)

21606374871245433044…43668094736177077399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.160 × 10⁹⁴(95-digit number)

21606374871245433044…43668094736177077401

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

4.321 × 10⁹⁴(95-digit number)

43212749742490866088…87336189472354154799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.321 × 10⁹⁴(95-digit number)

43212749742490866088…87336189472354154801

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

8.642 × 10⁹⁴(95-digit number)

86425499484981732177…74672378944708309599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

8.642 × 10⁹⁴(95-digit number)

86425499484981732177…74672378944708309601

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

1.728 × 10⁹⁵(96-digit number)

17285099896996346435…49344757889416619199

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.728 × 10⁹⁵(96-digit number)

17285099896996346435…49344757889416619201

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 #251,961

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