Block #157,721

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/9/2013, 8:07:57 PM · Difficulty 9.8690 · 6,636,327 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

9981f2f983a3e1d748e7fead0c446c81c0323dde80dc7e3fa36656a85bee2d6d

View on Chainz ↗

Height

#157,721

Difficulty

9.869007

Transactions

Size

199 B

Version

Bits

09de773c

Nonce

100,390

Timestamp

9/9/2013, 8:07:57 PM

Confirmations

6,636,327

Merkle Root

775a4525d505cec8d4acc85c184d5fc30016b36adb2416d03d141e52b38e4a5d

Transactions (1)

Coinbase775a4525d505ce…141e52b38e4a5d

1 in → 1 out10.2500 XPM109 B

ANNP26d7…c8ABLJ7e

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.

5.967 × 10⁹⁴(95-digit number)

59671898140044616720…51187889595925416959

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

5.967 × 10⁹⁴(95-digit number)

59671898140044616720…51187889595925416959

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.967 × 10⁹⁴(95-digit number)

59671898140044616720…51187889595925416961

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

1.193 × 10⁹⁵(96-digit number)

11934379628008923344…02375779191850833919

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.193 × 10⁹⁵(96-digit number)

11934379628008923344…02375779191850833921

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

2.386 × 10⁹⁵(96-digit number)

23868759256017846688…04751558383701667839

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.386 × 10⁹⁵(96-digit number)

23868759256017846688…04751558383701667841

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

4.773 × 10⁹⁵(96-digit number)

47737518512035693376…09503116767403335679

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.773 × 10⁹⁵(96-digit number)

47737518512035693376…09503116767403335681

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

9.547 × 10⁹⁵(96-digit number)

95475037024071386752…19006233534806671359

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