Block #261,759

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/16/2013, 3:10:39 AM · Difficulty 9.9709 · 6,554,916 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

daf18e7807bfe14d0084928b0cba6dd0df3d7c77d52d358d29f22c500dbc78e2

View on Chainz ↗

Height

#261,759

Difficulty

9.970858

Transactions

Size

2.04 KB

Version

Bits

09f88a22

Nonce

57,845

Timestamp

11/16/2013, 3:10:39 AM

Confirmations

6,554,916

Mined by

⛏️ aR9AYTdjuNAxqf6ZfbVvCfWQo5B5gth8FbAeh

Merkle Root

d96ae76ac5ca51e1e70512952d0f8602e02a248eccba3ae143d46358dffc6178

Transactions (1)

Coinbased96ae76ac5ca51…d46358dffc6178

1 in → 57 out10.0200 XPM1.95 KB

AYTdjuNA…th8FbAeh APH8rV6F…heb1HF2Z AQtzUSwq…htAuF3yC APZy8y6E…QZaGQjfp ANHbaxZp…XjnHCJJs

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.

9.613 × 10⁹³(94-digit number)

96132697382020804657…29053584757381018879

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

9.613 × 10⁹³(94-digit number)

96132697382020804657…29053584757381018879

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

9.613 × 10⁹³(94-digit number)

96132697382020804657…29053584757381018881

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

1.922 × 10⁹⁴(95-digit number)

19226539476404160931…58107169514762037759

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.922 × 10⁹⁴(95-digit number)

19226539476404160931…58107169514762037761

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

3.845 × 10⁹⁴(95-digit number)

38453078952808321862…16214339029524075519

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.845 × 10⁹⁴(95-digit number)

38453078952808321862…16214339029524075521

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

7.690 × 10⁹⁴(95-digit number)

76906157905616643725…32428678059048151039

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.690 × 10⁹⁴(95-digit number)

76906157905616643725…32428678059048151041

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

1.538 × 10⁹⁵(96-digit number)

15381231581123328745…64857356118096302079

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.538 × 10⁹⁵(96-digit number)

15381231581123328745…64857356118096302081

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

Block #261,759

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