Block #313,499

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/15/2013, 12:22:11 PM · Difficulty 10.0085 · 6,482,620 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

eff3c24725a6d2232d576e161ed61af03d1ff9f908d042516ee3a84d316b4c43

View on Chainz ↗

Height

#313,499

Difficulty

10.008496

Transactions

Size

1.11 KB

Version

Bits

0a022ccd

Nonce

210,658

Timestamp

12/15/2013, 12:22:11 PM

Confirmations

6,482,620

Mined by

⛏️ aRAbtgNzNbw1hJh3uoaBwXxdpmiY9Atvu81w

Merkle Root

f07a1c339c3a346f7e88ea57f8aa1fc55c0ce953ffea0f5ad48df05978bcf22f

Transactions (1)

Coinbasef07a1c339c3a34…8df05978bcf22f

1 in → 29 out9.9500 XPM1.02 KB

AbtgNzNb…9Atvu81w Aac9Hjdg…P4ktypc3 AYcLTeXR…bscgPops Ad9pSzHt…cpJ3y2zz AJzWx2VD…j4ZSMit5

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.

3.495 × 10¹⁰⁰(101-digit number)

34953118100649352151…64881022073871999999

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

3.495 × 10¹⁰⁰(101-digit number)

34953118100649352151…64881022073871999999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.495 × 10¹⁰⁰(101-digit number)

34953118100649352151…64881022073872000001

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

6.990 × 10¹⁰⁰(101-digit number)

69906236201298704303…29762044147743999999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.990 × 10¹⁰⁰(101-digit number)

69906236201298704303…29762044147744000001

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

1.398 × 10¹⁰¹(102-digit number)

13981247240259740860…59524088295487999999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.398 × 10¹⁰¹(102-digit number)

13981247240259740860…59524088295488000001

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

2.796 × 10¹⁰¹(102-digit number)

27962494480519481721…19048176590975999999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.796 × 10¹⁰¹(102-digit number)

27962494480519481721…19048176590976000001

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

5.592 × 10¹⁰¹(102-digit number)

55924988961038963442…38096353181951999999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

5.592 × 10¹⁰¹(102-digit number)

55924988961038963442…38096353181952000001

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