Block #140,657

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 8/29/2013, 7:13:33 PM · Difficulty 9.8341 · 6,674,179 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e1c3af39d537e102f1ff25715381d4e0b74967f48fb1a7dd8d4f911f250e888e

View on Chainz ↗

Height

#140,657

Difficulty

9.834068

Transactions

Size

1.00 KB

Version

Bits

09d58575

Nonce

417,611

Timestamp

8/29/2013, 7:13:33 PM

Confirmations

6,674,179

Mined by

⛏️ P2SHAbGZum2dsWmKHeCJmESx4VYuDe44MDDShz

Merkle Root

a93972cbac426f9153e3fec686bc892fb07876a8e373e5a302deb1d748bbfd2d

Transactions (4)

Coinbasee205397b096b99…44b656db891cb7

1 in → 1 out10.3500 XPM109 B

AbGZum2d…44MDDShz

8cbf00d5cae026…23a622b46878b8

1 in → 2 out10.3400 XPM226 B

AGcXCdtw…7TU8Y5ye AYKVeNA9…BbJrTDhi

5ee03f58bdb6d2…60da223da31397

2 in → 2 out10.5608 XPM374 B

AaamjEDB…UqchbkWj APiDuhZM…raa4bKzM

5ab8e7eb7d0afe…1dfb2aaa69b68b

1 in → 2 out5.7805 XPM226 B

AUT1qxTx…t5B3qd8Z ALiDLdZy…gXsHQFpR

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.

8.019 × 10⁹⁴(95-digit number)

80198876530824312985…54200069850551814469

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

8.019 × 10⁹⁴(95-digit number)

80198876530824312985…54200069850551814469

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.019 × 10⁹⁴(95-digit number)

80198876530824312985…54200069850551814471

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

1.603 × 10⁹⁵(96-digit number)

16039775306164862597…08400139701103628939

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.603 × 10⁹⁵(96-digit number)

16039775306164862597…08400139701103628941

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

3.207 × 10⁹⁵(96-digit number)

32079550612329725194…16800279402207257879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.207 × 10⁹⁵(96-digit number)

32079550612329725194…16800279402207257881

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

6.415 × 10⁹⁵(96-digit number)

64159101224659450388…33600558804414515759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.415 × 10⁹⁵(96-digit number)

64159101224659450388…33600558804414515761

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

1.283 × 10⁹⁶(97-digit number)

12831820244931890077…67201117608829031519

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 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 9

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

Block #140,657

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