Block #167,751

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/16/2013, 8:15:05 PM · Difficulty 9.8681 · 6,627,988 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

d5e895fb6e47f27cc21acbc615231ece765939df954506bae451cd1627049dbb

View on Chainz ↗

Height

#167,751

Difficulty

9.868056

Transactions

Size

6.42 KB

Version

Bits

09de38e8

Nonce

858,571

Timestamp

9/16/2013, 8:15:05 PM

Confirmations

6,627,988

Mined by

⛏️ P2SHAXYcP8SHsYpqq9uuQTxUHcX8K9Fi18gYAF

Merkle Root

1c404b0dc6145850a4c3472ead49a8eba56ef235a22e84b0bf4508d67b118c92

Transactions (3)

Coinbase19fef80549d415…2dbc9c92a91db2

1 in → 1 out10.3200 XPM109 B

AXYcP8SH…Fi18gYAF

99d60fd67c82bb…40077af5a7afa4

5 in → 2 out42.2561 XPM818 B

AcUCgPX6…D3MC4D2d AX13Suqs…5uhsumtL

70411f388738f2…d99c81edc7fd64

37 in → 2 out223.0826 XPM5.42 KB

AV6URp9C…SYeCgYBK AYy8jR6C…vcoi2fmB

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.

4.372 × 10⁸⁹(90-digit number)

43729816967226707190…71403578357409331839

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

4.372 × 10⁸⁹(90-digit number)

43729816967226707190…71403578357409331839

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.372 × 10⁸⁹(90-digit number)

43729816967226707190…71403578357409331841

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

8.745 × 10⁸⁹(90-digit number)

87459633934453414380…42807156714818663679

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.745 × 10⁸⁹(90-digit number)

87459633934453414380…42807156714818663681

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

1.749 × 10⁹⁰(91-digit number)

17491926786890682876…85614313429637327359

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.749 × 10⁹⁰(91-digit number)

17491926786890682876…85614313429637327361

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

3.498 × 10⁹⁰(91-digit number)

34983853573781365752…71228626859274654719

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.498 × 10⁹⁰(91-digit number)

34983853573781365752…71228626859274654721

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

6.996 × 10⁹⁰(91-digit number)

69967707147562731504…42457253718549309439

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