Block #691,218

TWNLength 12★★★★☆

Bi-Twin Chain · Discovered 8/24/2014, 9:06:20 PM · Difficulty 10.9550 · 6,119,634 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

61a991da134b4b9c1424c31893dc80a0dfcd3c99978859736e2d5729e6836d73

View on Chainz ↗

Height

#691,218

Difficulty

10.955048

Transactions

Size

953 B

Version

Bits

0af47e05

Nonce

1,347,808,234

Timestamp

8/24/2014, 9:06:20 PM

Confirmations

6,119,634

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

7e7bb7e9b040bc097c27c07f45e2a122e8b34b7402dbf4e047f3371395667417

Transactions (3)

Coinbase7188d06172ffe0…2e01717c9ffceb

1 in → 1 out8.3400 XPM116 B

ANSXnLY2…Sd25TjkD

7c711470438018…046aadc1f02d5d

3 in → 2 out2.9226 XPM521 B

AeKNrjNj…1NEq95Ta AYt5grtd…X2DHMmwW

988ee607e93109…6ab9a62ba346e5

1 in → 2 out7.2650 XPM225 B

ANDsmKk6…S8xXEUa7 AJTsLJ89…rErdMt2e

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.

6.361 × 10⁹⁶(97-digit number)

63615360313721081193…30624904985977126399

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

6.361 × 10⁹⁶(97-digit number)

63615360313721081193…30624904985977126399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.361 × 10⁹⁶(97-digit number)

63615360313721081193…30624904985977126401

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

1.272 × 10⁹⁷(98-digit number)

12723072062744216238…61249809971954252799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.272 × 10⁹⁷(98-digit number)

12723072062744216238…61249809971954252801

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

2.544 × 10⁹⁷(98-digit number)

25446144125488432477…22499619943908505599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.544 × 10⁹⁷(98-digit number)

25446144125488432477…22499619943908505601

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

5.089 × 10⁹⁷(98-digit number)

50892288250976864954…44999239887817011199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.089 × 10⁹⁷(98-digit number)

50892288250976864954…44999239887817011201

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

1.017 × 10⁹⁸(99-digit number)

10178457650195372990…89998479775634022399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.017 × 10⁹⁸(99-digit number)

10178457650195372990…89998479775634022401

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

2.035 × 10⁹⁸(99-digit number)

20356915300390745981…79996959551268044799

Verify on FactorDB ↗Wolfram Alpha ↗

2^5 × origin + 1

2.035 × 10⁹⁸(99-digit number)

20356915300390745981…79996959551268044801

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^5 × origin + 1 − 2^5 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 12 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

ExceptionalChain length 12

Around 1 in 10,000 blocks. A significant mathematical achievement.

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 #691,218

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