Block #311,297

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 12:28:17 PM · Difficulty 9.9954 · 6,495,670 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3a291325f24b3de9aca4d6f4c87a10e41a7e6006f2bff12d52aebcb5d3080943

View on Chainz ↗

Height

#311,297

Difficulty

9.995419

Transactions

Size

6.51 KB

Version

Bits

09fed3cc

Nonce

73,192

Timestamp

12/14/2013, 12:28:17 PM

Confirmations

6,495,670

Mined by

⛏️ aRNdAac9Hjdgnw4T4L2csqLecJsmZjP4ktypc3

Merkle Root

1b13cbe19882d7b7485c80333d09433565ef22cac96b9aa55d9178f726fef751

Transactions (4)

Coinbase29bfcbf7400978…c91e2aa19e792f

1 in → 29 out9.9700 XPM1.02 KB

Aac9Hjdg…P4ktypc3 AbtgNzNb…9Atvu81w Ac4BCsHq…ZY22nyhE AcZViXQA…miP6vJYx Ad9pSzHt…cpJ3y2zz

bf88fb9e417c53…b94a0871517272

34 in → 1 out11.1784 XPM4.96 KB

AeDyRSkQ…8oncMSux

f0a3bf535aef18…fb3829eb94496d

1 in → 2 out810.2240 XPM227 B

AMRWq8WL…BapFcwsV APrDRvQo…VvQSZcjF

9ac6fccfe89a34…1d2788a22716a1

1 in → 2 out8.9544 XPM226 B

ASRkkY4V…fZo4hJq2 AYJocoLi…V4exF3qS

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.

6.676 × 10⁹⁹(100-digit number)

66768175957713666643…04659626744702609919

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

6.676 × 10⁹⁹(100-digit number)

66768175957713666643…04659626744702609919

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.676 × 10⁹⁹(100-digit number)

66768175957713666643…04659626744702609921

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

1.335 × 10¹⁰⁰(101-digit number)

13353635191542733328…09319253489405219839

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.335 × 10¹⁰⁰(101-digit number)

13353635191542733328…09319253489405219841

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

2.670 × 10¹⁰⁰(101-digit number)

26707270383085466657…18638506978810439679

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.670 × 10¹⁰⁰(101-digit number)

26707270383085466657…18638506978810439681

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

5.341 × 10¹⁰⁰(101-digit number)

53414540766170933314…37277013957620879359

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.341 × 10¹⁰⁰(101-digit number)

53414540766170933314…37277013957620879361

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

1.068 × 10¹⁰¹(102-digit number)

10682908153234186662…74554027915241758719

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 #311,297

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