Block #157,085

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/9/2013, 9:40:22 AM · Difficulty 9.8688 · 6,653,975 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

7db0f4b6f73cc4def840e8a9c879614106357cbbcf2474efd735ee0bea083cb3

View on Chainz ↗

Height

#157,085

Difficulty

9.868843

Transactions

Size

812 B

Version

Bits

09de6c85

Nonce

438,618

Timestamp

9/9/2013, 9:40:22 AM

Confirmations

6,653,975

Mined by

⛏️ P2SHAXcp4kJSJMTRmHejHzyoCze5ogAzAKPB9Y

Merkle Root

4e0c6e7ccc46493197b5910d1d51a5106a01fefc8d49df53db6034d80526e967

Transactions (2)

Coinbasef78ad7df62a00d…f1347b04054248

1 in → 1 out10.2600 XPM109 B

AXcp4kJS…AzAKPB9Y

328beac91b0834…3eb706fe008544

5 in → 1 out51.3700 XPM614 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.

6.896 × 10⁹¹(92-digit number)

68969471430302822045…90958558983476349699

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

6.896 × 10⁹¹(92-digit number)

68969471430302822045…90958558983476349699

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.896 × 10⁹¹(92-digit number)

68969471430302822045…90958558983476349701

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

1.379 × 10⁹²(93-digit number)

13793894286060564409…81917117966952699399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.379 × 10⁹²(93-digit number)

13793894286060564409…81917117966952699401

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

2.758 × 10⁹²(93-digit number)

27587788572121128818…63834235933905398799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.758 × 10⁹²(93-digit number)

27587788572121128818…63834235933905398801

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

5.517 × 10⁹²(93-digit number)

55175577144242257636…27668471867810797599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.517 × 10⁹²(93-digit number)

55175577144242257636…27668471867810797601

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

1.103 × 10⁹³(94-digit number)

11035115428848451527…55336943735621595199

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 #157,085

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