Block #151,698

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/5/2013, 8:20:12 PM · Difficulty 9.8615 · 6,657,438 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

da33b10e5a9d600e8136fe014d973483756335385a45cdd446d4c72646a774e3

View on Chainz ↗

Height

#151,698

Difficulty

9.861462

Transactions

Size

740 B

Version

Bits

09dc88c0

Nonce

27,554

Timestamp

9/5/2013, 8:20:12 PM

Confirmations

6,657,438

Mined by

⛏️ P2SHAaEaYYnPHwN5nsaWg4Ph9o3KviGEYib9go

Merkle Root

9c1fc75401b8dde7b3ea88b24295194e7dd33f5d09c23e1dc9443cf83097711c

Transactions (3)

Coinbase22a0ca02aa801c…b683877162d5c4

1 in → 1 out10.2900 XPM109 B

AaEaYYnP…GEYib9go

9b499a7102ff56…2c963c530962d8

2 in → 1 out20.5900 XPM271 B

Abg9b4kX…im9CLs7g

e437a3db2b6d6e…41e703132f5364

2 in → 1 out20.5800 XPM272 B

Abg9b4kX…im9CLs7g

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.

1.672 × 10⁹⁰(91-digit number)

16729255984945256646…79730751459094582159

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

1.672 × 10⁹⁰(91-digit number)

16729255984945256646…79730751459094582159

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.672 × 10⁹⁰(91-digit number)

16729255984945256646…79730751459094582161

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

3.345 × 10⁹⁰(91-digit number)

33458511969890513293…59461502918189164319

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.345 × 10⁹⁰(91-digit number)

33458511969890513293…59461502918189164321

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

6.691 × 10⁹⁰(91-digit number)

66917023939781026587…18923005836378328639

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.691 × 10⁹⁰(91-digit number)

66917023939781026587…18923005836378328641

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

1.338 × 10⁹¹(92-digit number)

13383404787956205317…37846011672756657279

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.338 × 10⁹¹(92-digit number)

13383404787956205317…37846011672756657281

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

2.676 × 10⁹¹(92-digit number)

26766809575912410634…75692023345513314559

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 #151,698

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