Block #291,557

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/3/2013, 6:16:38 AM · Difficulty 9.9899 · 6,507,973 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

fff116320b10c5a6a76d75ee80ac4e61e6bd9a29301a49f984e37cec3fdaeb9d

View on Chainz ↗

Height

#291,557

Difficulty

9.989899

Transactions

Size

1.61 KB

Version

Bits

09fd69fd

Nonce

2,880

Timestamp

12/3/2013, 6:16:38 AM

Confirmations

6,507,973

Mined by

⛏️ raRw@AAYCeLhqBJ84jFrKumsmd4mf6paeqGM6KK1

Merkle Root

75af0bc567fa2712c6b07f78f96e28167932f1e320f9f76a25919f5b7bd697c7

Transactions (4)

Coinbased476360abbc11a…c73181523e7891

1 in → 1 out10.0100 XPM97 B

AYCeLhqB…eqGM6KK1

91491ac7d4cacf…d500d25e1051ec

4 in → 10 out33.7613 XPM841 B

ALanGMzb…mpKXHzz4 AXdpwESF…xST8zfa6 AW51412z…8H4N5txp AeKNrjNj…1NEq95Ta ALkuF5Je…KBUB19wB

c961b8b7bd6aef…531b8957f29953

1 in → 7 out19.5775 XPM396 B

AGxYGYHa…FRp3G4Gh AUJR8wsP…y9EiREVQ AP51GjXg…NhL4hYQz AHqGjrnz…i6FobY9E ARRBNu1j…CfVmGyj7

b5c20e97ba3a53…972b2e924fc933

1 in → 2 out1.6310 XPM225 B

AcDKAHmJ…ZGLUf4rV ANG8wpdK…BaAfRo75

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.

2.280 × 10⁹⁸(99-digit number)

22806702171646943781…75473092785163220499

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

2.280 × 10⁹⁸(99-digit number)

22806702171646943781…75473092785163220499

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.280 × 10⁹⁸(99-digit number)

22806702171646943781…75473092785163220501

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

4.561 × 10⁹⁸(99-digit number)

45613404343293887562…50946185570326440999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.561 × 10⁹⁸(99-digit number)

45613404343293887562…50946185570326441001

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

9.122 × 10⁹⁸(99-digit number)

91226808686587775124…01892371140652881999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.122 × 10⁹⁸(99-digit number)

91226808686587775124…01892371140652882001

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

1.824 × 10⁹⁹(100-digit number)

18245361737317555024…03784742281305763999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.824 × 10⁹⁹(100-digit number)

18245361737317555024…03784742281305764001

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

3.649 × 10⁹⁹(100-digit number)

36490723474635110049…07569484562611527999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.649 × 10⁹⁹(100-digit number)

36490723474635110049…07569484562611528001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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