Block #857,621

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/17/2014, 11:36:57 PM · Difficulty 10.9674 · 5,985,284 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

66ff75b26fc4db3ba274ebea81a9a2af9ac4ffae1460382aed94a8226624f29a

View on Chainz ↗

Height

#857,621

Difficulty

10.967428

Transactions

Size

883 B

Version

Bits

0af7a960

Nonce

413,501,519

Timestamp

12/17/2014, 11:36:57 PM

Confirmations

5,985,284

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

51e1b1efbe13b11124d4f60340ee66aa7905ffb6dbd30ce0d1466a343f399c19

Transactions (4)

Coinbaseddb03287e9bd65…a7a4be1e920985

1 in → 1 out8.3450 XPM116 B

ANSXnLY2…Sd25TjkD

bfc4a242bf3a53…ee2967e799b134

1 in → 2 out0.1070 XPM225 B

AQc3gUmj…Uqgyobyd AYTY9VX2…gw7hGBWR

8961b17658f380…0d000c74c3449d

1 in → 2 out0.0582 XPM225 B

AMCNhSNd…VVwPoKyt ARmX2MgH…yjCaN722

f523ce637d99df…a67ece84cae632

1 in → 2 out0.0451 XPM227 B

AS2rdA4C…upbhg5EW AJ7nztGJ…5M3tW3Xw

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.

2.326 × 10⁹⁴(95-digit number)

23266587445383309134…67888836454236159079

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

2.326 × 10⁹⁴(95-digit number)

23266587445383309134…67888836454236159079

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.326 × 10⁹⁴(95-digit number)

23266587445383309134…67888836454236159081

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

4.653 × 10⁹⁴(95-digit number)

46533174890766618268…35777672908472318159

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.653 × 10⁹⁴(95-digit number)

46533174890766618268…35777672908472318161

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

9.306 × 10⁹⁴(95-digit number)

93066349781533236536…71555345816944636319

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.306 × 10⁹⁴(95-digit number)

93066349781533236536…71555345816944636321

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

1.861 × 10⁹⁵(96-digit number)

18613269956306647307…43110691633889272639

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.861 × 10⁹⁵(96-digit number)

18613269956306647307…43110691633889272641

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

3.722 × 10⁹⁵(96-digit number)

37226539912613294614…86221383267778545279

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.722 × 10⁹⁵(96-digit number)

37226539912613294614…86221383267778545281

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 #857,621

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