Block #829,306

TWNLength 11★★★☆☆

Bi-Twin Chain · Discovered 11/26/2014, 5:29:44 PM · Difficulty 10.9786 · 5,977,602 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e9273c8f27142fb72021831f0edb65845bc21f78ce01ab4196085d79fb381f6a

View on Chainz ↗

Height

#829,306

Difficulty

10.978637

Transactions

Size

887 B

Version

Bits

0afa87ed

Nonce

570,656,671

Timestamp

11/26/2014, 5:29:44 PM

Confirmations

5,977,602

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

9b49c75998c77eac51495777d81d61b731f8ead61dc9e2df6c9af55125e48ac4

Transactions (4)

Coinbasea08c1b366b0007…8777c9827ea5b5

1 in → 1 out8.3100 XPM116 B

ANSXnLY2…Sd25TjkD

99ec28aab7c0fe…13efb806f0af09

1 in → 2 out8.2800 XPM227 B

AYJf3ru3…Z23LUU8h ALNQaD7f…2g74yYvV

f7f42bb6c3512e…35b0387362daf8

1 in → 2 out8.2800 XPM226 B

AMoJsji6…a13Bp22Q AKdRZjkg…KcZtsQEk

8d9a7736f65b4b…a4b9ad1a881591

1 in → 2 out6.4763 XPM227 B

AUjF8v7i…hR6AjNRh AcsaQZN3…zJtdvnFf

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.

5.151 × 10⁹⁷(98-digit number)

51519766054134574383…47599361264020172799

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

5.151 × 10⁹⁷(98-digit number)

51519766054134574383…47599361264020172799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.151 × 10⁹⁷(98-digit number)

51519766054134574383…47599361264020172801

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

1.030 × 10⁹⁸(99-digit number)

10303953210826914876…95198722528040345599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.030 × 10⁹⁸(99-digit number)

10303953210826914876…95198722528040345601

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

2.060 × 10⁹⁸(99-digit number)

20607906421653829753…90397445056080691199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.060 × 10⁹⁸(99-digit number)

20607906421653829753…90397445056080691201

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

4.121 × 10⁹⁸(99-digit number)

41215812843307659506…80794890112161382399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.121 × 10⁹⁸(99-digit number)

41215812843307659506…80794890112161382401

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

8.243 × 10⁹⁸(99-digit number)

82431625686615319013…61589780224322764799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.243 × 10⁹⁸(99-digit number)

82431625686615319013…61589780224322764801

Verify on FactorDB ↗Wolfram Alpha ↗

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

Level 5 — Twin Prime Pair (2^5 × origin ± 1)

2^5 × origin − 1

1.648 × 10⁹⁹(100-digit number)

16486325137323063802…23179560448645529599

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 11 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

RareChain length 11

Approximately 1 in 1,000 blocks. Noteworthy discoveries.

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 #829,306

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