Block #213,813

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/17/2013, 2:23:33 AM · Difficulty 9.9224 · 6,613,363 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e526d02160aab231a1b75fee4b32ba7938cb9a32afdc7f3aa39212b82c577b9c

View on Chainz ↗

Height

#213,813

Difficulty

9.922436

Transactions

Size

962 B

Version

Bits

09ec24c7

Nonce

2,395

Timestamp

10/17/2013, 2:23:33 AM

Confirmations

6,613,363

Mined by

⛏️ P2SHAbJUC2oSrZY84nuu7WUoHPgJjAcpKam6LK

Merkle Root

0fd3e0f3080c685dc526a4f6bd713b873af5dd92b1aebe224bbd20745550db74

Transactions (3)

Coinbased57cd268b5790a…37ed3d58779b04

1 in → 1 out10.1600 XPM109 B

AbJUC2oS…cpKam6LK

8e96e50a061c57…683ccb868c6a72

4 in → 2 out40.6831 XPM534 B

AZZPbi1u…YhPG9rbe AN6FXaNm…B8MSxG4J

d2f3df9057ba7d…bb28dc85a58d08

1 in → 2 out7.3427 XPM226 B

Ab3dSvM7…Qoxxb9tp Aeqqx1nD…DKqjMBRQ

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.

2.019 × 10¹⁰¹(102-digit number)

20192188444281243813…82873252152673322239

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

2.019 × 10¹⁰¹(102-digit number)

20192188444281243813…82873252152673322239

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.019 × 10¹⁰¹(102-digit number)

20192188444281243813…82873252152673322241

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

4.038 × 10¹⁰¹(102-digit number)

40384376888562487627…65746504305346644479

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.038 × 10¹⁰¹(102-digit number)

40384376888562487627…65746504305346644481

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

8.076 × 10¹⁰¹(102-digit number)

80768753777124975255…31493008610693288959

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.076 × 10¹⁰¹(102-digit number)

80768753777124975255…31493008610693288961

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

1.615 × 10¹⁰²(103-digit number)

16153750755424995051…62986017221386577919

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.615 × 10¹⁰²(103-digit number)

16153750755424995051…62986017221386577921

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

3.230 × 10¹⁰²(103-digit number)

32307501510849990102…25972034442773155839

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 #213,813

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