Block #513,165

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 4/27/2014, 8:26:32 AM · Difficulty 10.8350 · 6,281,418 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

0c8fb022445ebeaf74596a08ecb9d0b1a0857cc1c2c4cc3c599381971b8171bf

View on Chainz ↗

Height

#513,165

Difficulty

10.834977

Transactions

Size

827 B

Version

Bits

0ad5c10a

Nonce

183,942

Timestamp

4/27/2014, 8:26:32 AM

Confirmations

6,281,418

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

416e950e815558757a9cb89e49d86601a233aa20561c80d90436c46e3b2328a8

Transactions (2)

Coinbase10c5a3e4b65ef5…eb53e20b8d8af1

1 in → 1 out8.5100 XPM110 B

AeKNrjNj…1NEq95Ta

7c9f878ab77c33…b818b14d334a4c

3 in → 7 out18.9678 XPM624 B

Abpc9caP…YUV7o3Vb AMeyeZ9L…gJoDemgf AZTpkAuM…a7i5sa6o AZoyjqWo…cTWTRwva AKc9Rmjx…Bir3N5mm

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.

1.561 × 10¹⁰²(103-digit number)

15612265125643008348…66517753617321390079

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

1.561 × 10¹⁰²(103-digit number)

15612265125643008348…66517753617321390079

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.561 × 10¹⁰²(103-digit number)

15612265125643008348…66517753617321390081

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

3.122 × 10¹⁰²(103-digit number)

31224530251286016696…33035507234642780159

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.122 × 10¹⁰²(103-digit number)

31224530251286016696…33035507234642780161

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

6.244 × 10¹⁰²(103-digit number)

62449060502572033393…66071014469285560319

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.244 × 10¹⁰²(103-digit number)

62449060502572033393…66071014469285560321

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

1.248 × 10¹⁰³(104-digit number)

12489812100514406678…32142028938571120639

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.248 × 10¹⁰³(104-digit number)

12489812100514406678…32142028938571120641

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

2.497 × 10¹⁰³(104-digit number)

24979624201028813357…64284057877142241279

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.497 × 10¹⁰³(104-digit number)

24979624201028813357…64284057877142241281

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