Block #315,706

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/16/2013, 3:52:09 PM · Difficulty 10.1123 · 6,509,986 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

d303afed468ff0418fcdcf8645aaef6beb110321dc71a325453a4c3dc06c252c

View on Chainz ↗

Height

#315,706

Difficulty

10.112331

Transactions

Size

2.14 KB

Version

Bits

0a1cc1ba

Nonce

11,083

Timestamp

12/16/2013, 3:52:09 PM

Confirmations

6,509,986

Mined by

⛏️ :aR!AQwRN8bEjE7sawFBq7N38V9niAV8PsTcY9

Merkle Root

cd3a1cacd8e1d08f998f967e25d2979e7037547bbf573f2ad4d8557774869e4d

Transactions (4)

Coinbase3a56d08dd979fa…c3c755d4c8f5d2

1 in → 27 out9.7600 XPM981 B

AQwRN8bE…V8PsTcY9 Aac9Hjdg…P4ktypc3 Ad9pSzHt…cpJ3y2zz AYtDvcGs…VuAcA7m7 Ae58nAEb…GFUA15fv

a61146b206c84a…c1fc75abfda190

4 in → 2 out14.0951 XPM670 B

ASVquHxC…DvoeLXY6 AXgmkZjS…dsm11AG7

a05716ac9f698a…e087daa3320957

1 in → 2 out6.6195 XPM226 B

AKauPgm1…5p9BgSVA AVyGxJhW…FgWfSq4s

2436efe90a13d7…70dbbd47c27dbb

1 in → 2 out5.8117 XPM227 B

AHXW8iGj…8V5oGVKS AGys8Jhb…pkaqzc2A

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.

2.993 × 10⁹⁴(95-digit number)

29938679892664261476…70065539178905030399

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

2.993 × 10⁹⁴(95-digit number)

29938679892664261476…70065539178905030399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.993 × 10⁹⁴(95-digit number)

29938679892664261476…70065539178905030401

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

5.987 × 10⁹⁴(95-digit number)

59877359785328522952…40131078357810060799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.987 × 10⁹⁴(95-digit number)

59877359785328522952…40131078357810060801

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

1.197 × 10⁹⁵(96-digit number)

11975471957065704590…80262156715620121599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.197 × 10⁹⁵(96-digit number)

11975471957065704590…80262156715620121601

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

2.395 × 10⁹⁵(96-digit number)

23950943914131409180…60524313431240243199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.395 × 10⁹⁵(96-digit number)

23950943914131409180…60524313431240243201

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

4.790 × 10⁹⁵(96-digit number)

47901887828262818361…21048626862480486399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.790 × 10⁹⁵(96-digit number)

47901887828262818361…21048626862480486401

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 #315,706

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