Block #316,398

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/17/2013, 1:23:55 AM · Difficulty 10.1335 · 6,478,045 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

c561c723e5c25fc68939fa3b4b1f5d0b7855dd27cd0e02f53d7bdabeb6969645

View on Chainz ↗

Height

#316,398

Difficulty

10.133548

Transactions

Size

1.05 KB

Version

Bits

0a223033

Nonce

3,634

Timestamp

12/17/2013, 1:23:55 AM

Confirmations

6,478,045

Mined by

⛏️ aRAd9pSzHtZ9ebPysWxo4VY8quTmcpJ3y2zz

Merkle Root

e65d79f4d4c1c4db19183eb4d11a962016455af72e976a35a478136417c0940a

Transactions (1)

Coinbasee65d79f4d4c1c4…78136417c0940a

1 in → 27 out9.7200 XPM981 B

Ad9pSzHt…cpJ3y2zz AYtDvcGs…VuAcA7m7 AbtgNzNb…9Atvu81w Aac9Hjdg…P4ktypc3 AM1wsdX3…NqVu85hE

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.

2.067 × 10¹⁰¹(102-digit number)

20674621832377323306…78052797506863922879

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

2.067 × 10¹⁰¹(102-digit number)

20674621832377323306…78052797506863922879

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.067 × 10¹⁰¹(102-digit number)

20674621832377323306…78052797506863922881

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

4.134 × 10¹⁰¹(102-digit number)

41349243664754646612…56105595013727845759

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.134 × 10¹⁰¹(102-digit number)

41349243664754646612…56105595013727845761

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

8.269 × 10¹⁰¹(102-digit number)

82698487329509293225…12211190027455691519

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.269 × 10¹⁰¹(102-digit number)

82698487329509293225…12211190027455691521

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

1.653 × 10¹⁰²(103-digit number)

16539697465901858645…24422380054911383039

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.653 × 10¹⁰²(103-digit number)

16539697465901858645…24422380054911383041

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

3.307 × 10¹⁰²(103-digit number)

33079394931803717290…48844760109822766079

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.307 × 10¹⁰²(103-digit number)

33079394931803717290…48844760109822766081

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