Block #10,513

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/11/2013, 3:18:35 AM · Difficulty 7.6754 · 6,792,146 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

15d66729e5f02917d36b504c02d70a86fa1cc7a88932efe572a9e16f3007f053

View on Chainz ↗

Height

#10,513

Difficulty

7.675432

Transactions

Size

654 B

Version

Bits

07ace91c

Nonce

578

Timestamp

7/11/2013, 3:18:35 AM

Confirmations

6,792,146

Mined by

⛏️ P2SHATN2gTAz1dzd2JJe16voWBC6s8fwgpStqr

Merkle Root

ffb035eca81fd2da48983794da93a97518287fe9058cc1690a68778ecbf2f233

Transactions (2)

Coinbased6dee54a65db3e…80d0b34a9009d8

1 in → 1 out16.9600 XPM108 B

ATN2gTAz…fwgpStqr

19bd93b05b9d1e…08177c43fb4ddd

3 in → 2 out56.9700 XPM455 B

ATxihnQj…UcwvsSkn Ab413rus…pRb5fo8F

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.

5.278 × 10⁹⁶(97-digit number)

52788780077890734847…68484304286920514639

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

5.278 × 10⁹⁶(97-digit number)

52788780077890734847…68484304286920514639

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.278 × 10⁹⁶(97-digit number)

52788780077890734847…68484304286920514641

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

1.055 × 10⁹⁷(98-digit number)

10557756015578146969…36968608573841029279

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.055 × 10⁹⁷(98-digit number)

10557756015578146969…36968608573841029281

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

2.111 × 10⁹⁷(98-digit number)

21115512031156293938…73937217147682058559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.111 × 10⁹⁷(98-digit number)

21115512031156293938…73937217147682058561

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

4.223 × 10⁹⁷(98-digit number)

42231024062312587877…47874434295364117119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.223 × 10⁹⁷(98-digit number)

42231024062312587877…47874434295364117121

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 8 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 8

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