Block #310,967

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 8:15:04 AM · Difficulty 9.9953 · 6,483,981 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

952f441afc09942ac7f6c7de0e74404d896a5fb65990c60d01b039e847587ff5

View on Chainz ↗

Height

#310,967

Difficulty

9.995338

Transactions

Size

1.74 KB

Version

Bits

09fece79

Nonce

48,480

Timestamp

12/14/2013, 8:15:04 AM

Confirmations

6,483,981

Mined by

⛏️ aRAPeshfYVKJ2qg4aRhC5FMjPmxg3PKcKMKH

Merkle Root

c021d219cb6c4264eda706e39f9de56676a3ae99f76867abaee9800a857b23f3

Transactions (4)

Coinbasecf82a1acafbe26…8dd84a4c386648

1 in → 28 out9.9700 XPM1015 B

APeshfYV…3PKcKMKH Aac9Hjdg…P4ktypc3 AbtgNzNb…9Atvu81w AZ2pPuKg…95SN2Yr7 AHCbk3sT…ddhvYDDU

3f78b98b7e53f4…97080760126282

1 in → 2 out2646.5727 XPM226 B

AZPzT7nq…JQTKK4oV AMRWq8WL…BapFcwsV

946b60e0f9c1a3…5e9b8217dc0ccc

1 in → 2 out6.4975 XPM225 B

AKvpqHm3…muEeCCpL AGjiJUqz…noYXu1sW

9980bdd748463a…6bb2c1f50b77a7

1 in → 2 out6.3085 XPM225 B

AWHqSsod…S114Gc1Y Abtsu2Tx…q88eHDy9

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.

8.953 × 10⁹⁵(96-digit number)

89539757052588333877…23115420370875604479

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

8.953 × 10⁹⁵(96-digit number)

89539757052588333877…23115420370875604479

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.953 × 10⁹⁵(96-digit number)

89539757052588333877…23115420370875604481

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

1.790 × 10⁹⁶(97-digit number)

17907951410517666775…46230840741751208959

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.790 × 10⁹⁶(97-digit number)

17907951410517666775…46230840741751208961

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

3.581 × 10⁹⁶(97-digit number)

35815902821035333551…92461681483502417919

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.581 × 10⁹⁶(97-digit number)

35815902821035333551…92461681483502417921

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

7.163 × 10⁹⁶(97-digit number)

71631805642070667102…84923362967004835839

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.163 × 10⁹⁶(97-digit number)

71631805642070667102…84923362967004835841

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

1.432 × 10⁹⁷(98-digit number)

14326361128414133420…69846725934009671679

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