Block #39,659

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/14/2013, 1:50:05 PM · Difficulty 8.3456 · 6,775,261 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

67324de86c98f8beefd6e52cdf0983815c4d9fb61fd26ec1bd25a0deef3d3b69

View on Chainz ↗

Height

#39,659

Difficulty

8.345555

Transactions

Size

362 B

Version

Bits

08587646

Nonce

314

Timestamp

7/14/2013, 1:50:05 PM

Confirmations

6,775,261

Mined by

⛏️ P2SHAG3YzYknby2UPyoshZTYqbpnqA815RUkjE

Merkle Root

190791cafcf02d5d3c29015970cc6b23d6c63266acb0750098c4722368a94fdc

Transactions (2)

Coinbase0ff0c135365b2a…a678a3565c3fbc

1 in → 1 out14.3500 XPM110 B

AG3YzYkn…815RUkjE

f5fb1c1e8ab90e…7a29ce353811a5

1 in → 1 out15.6300 XPM157 B

ATaF1JyB…Lo75UW58

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.

2.181 × 10¹⁰⁶(107-digit number)

21811490312944655399…62908557628340424939

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

2.181 × 10¹⁰⁶(107-digit number)

21811490312944655399…62908557628340424939

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.181 × 10¹⁰⁶(107-digit number)

21811490312944655399…62908557628340424941

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

4.362 × 10¹⁰⁶(107-digit number)

43622980625889310799…25817115256680849879

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.362 × 10¹⁰⁶(107-digit number)

43622980625889310799…25817115256680849881

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

8.724 × 10¹⁰⁶(107-digit number)

87245961251778621598…51634230513361699759

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.724 × 10¹⁰⁶(107-digit number)

87245961251778621598…51634230513361699761

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

1.744 × 10¹⁰⁷(108-digit number)

17449192250355724319…03268461026723399519

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.744 × 10¹⁰⁷(108-digit number)

17449192250355724319…03268461026723399521

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 8 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 8

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 #39,659

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