Block #314,513

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/16/2013, 12:16:23 AM · Difficulty 10.0663 · 6,479,561 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b57e01111ecb6de9b4419d7250e35f170365706f198cb9bc3d84ff19efdfdcf2

View on Chainz ↗

Height

#314,513

Difficulty

10.066283

Transactions

Size

1.90 KB

Version

Bits

0a10f7eb

Nonce

110,778

Timestamp

12/16/2013, 12:16:23 AM

Confirmations

6,479,561

Merkle Root

fad20c5847fc40dfc6521f68ce7fbdcdfcd1b42372cf99e8f74a94a1662a6da7

Transactions (2)

Coinbase56c6459f484efd…6da71279e51a07

1 in → 1 out9.8700 XPM110 B

AeKNrjNj…1NEq95Ta

8721333566a846…7dc2eb674ebcfc

8 in → 23 out75.4533 XPM1.70 KB

AKF6HtYj…qBVvCufz AL9eunh9…93skgvXX AcK1VmKm…2EcxV3uw APT4WL4a…m8hRQBQU ALAk6Jx9…MWDjHKLd

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.

2.972 × 10¹⁰⁰(101-digit number)

29723616138540130233…25654731101786076159

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

2.972 × 10¹⁰⁰(101-digit number)

29723616138540130233…25654731101786076159

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.972 × 10¹⁰⁰(101-digit number)

29723616138540130233…25654731101786076161

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

5.944 × 10¹⁰⁰(101-digit number)

59447232277080260466…51309462203572152319

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.944 × 10¹⁰⁰(101-digit number)

59447232277080260466…51309462203572152321

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

1.188 × 10¹⁰¹(102-digit number)

11889446455416052093…02618924407144304639

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.188 × 10¹⁰¹(102-digit number)

11889446455416052093…02618924407144304641

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

2.377 × 10¹⁰¹(102-digit number)

23778892910832104186…05237848814288609279

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.377 × 10¹⁰¹(102-digit number)

23778892910832104186…05237848814288609281

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

4.755 × 10¹⁰¹(102-digit number)

47557785821664208373…10475697628577218559

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.755 × 10¹⁰¹(102-digit number)

47557785821664208373…10475697628577218561

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