Block #164,057

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/14/2013, 10:33:07 AM · Difficulty 9.8616 · 6,631,517 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

29791daea512000d97969ff7af87c8042f03d5f0f16496ca8d79baebb4bdf747

View on Chainz ↗

Height

#164,057

Difficulty

9.861582

Transactions

Size

425 B

Version

Bits

09dc90a9

Nonce

36,624

Timestamp

9/14/2013, 10:33:07 AM

Confirmations

6,631,517

Mined by

⛏️ P2SHAYbFBdKztvJRWUEaa1hwf4Ak9pcf926Mct

Merkle Root

2bd148a3c3f16a855bd0a46ae25ff759d1e06e3bcc01f9772bac2f511291d51c

Transactions (2)

Coinbase6ae749940c527c…3c245d4c33af0a

1 in → 1 out10.2800 XPM109 B

AYbFBdKz…cf926Mct

0034fdec6d2192…160010a8ba8d10

1 in → 2 out2.0361 XPM226 B

AGnvPptD…Ts3omEe7 AYy6jDKp…aS5b1Ft6

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.

4.525 × 10⁹³(94-digit number)

45253335021428239223…25955800722344431359

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

4.525 × 10⁹³(94-digit number)

45253335021428239223…25955800722344431359

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.525 × 10⁹³(94-digit number)

45253335021428239223…25955800722344431361

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

9.050 × 10⁹³(94-digit number)

90506670042856478446…51911601444688862719

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

9.050 × 10⁹³(94-digit number)

90506670042856478446…51911601444688862721

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

1.810 × 10⁹⁴(95-digit number)

18101334008571295689…03823202889377725439

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.810 × 10⁹⁴(95-digit number)

18101334008571295689…03823202889377725441

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

3.620 × 10⁹⁴(95-digit number)

36202668017142591378…07646405778755450879

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.620 × 10⁹⁴(95-digit number)

36202668017142591378…07646405778755450881

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

7.240 × 10⁹⁴(95-digit number)

72405336034285182757…15292811557510901759

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