Block #310,531

TWNLength 11★★★☆☆

Bi-Twin Chain · Discovered 12/14/2013, 2:57:44 AM · Difficulty 9.9952 · 6,492,106 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e56599fc380c01ebd7d0666b198c46988dacc52729cc7d585854f742458da97f

View on Chainz ↗

Height

#310,531

Difficulty

9.995214

Transactions

Size

969 B

Version

Bits

09fec657

Nonce

88,456

Timestamp

12/14/2013, 2:57:44 AM

Confirmations

6,492,106

Mined by

⛏️ aRAXmS7tZuzGFK763sTons6zqW8k11YypHLP

Merkle Root

9d428f1da6d2455e77fc74cff4f84a90cd14336248d82f3fabde989277606ba3

Transactions (1)

Coinbase9d428f1da6d245…de989277606ba3

1 in → 24 out9.9870 XPM879 B

AXmS7tZu…11YypHLP Aac9Hjdg…P4ktypc3 Ad9pSzHt…cpJ3y2zz AayReikY…2HAC42fs AYcLTeXR…bscgPops

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.

7.776 × 10⁹³(94-digit number)

77767758459934243878…51172233756887734999

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

7.776 × 10⁹³(94-digit number)

77767758459934243878…51172233756887734999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.776 × 10⁹³(94-digit number)

77767758459934243878…51172233756887735001

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

1.555 × 10⁹⁴(95-digit number)

15553551691986848775…02344467513775469999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.555 × 10⁹⁴(95-digit number)

15553551691986848775…02344467513775470001

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

3.110 × 10⁹⁴(95-digit number)

31107103383973697551…04688935027550939999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.110 × 10⁹⁴(95-digit number)

31107103383973697551…04688935027550940001

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

6.221 × 10⁹⁴(95-digit number)

62214206767947395103…09377870055101879999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.221 × 10⁹⁴(95-digit number)

62214206767947395103…09377870055101880001

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

1.244 × 10⁹⁵(96-digit number)

12442841353589479020…18755740110203759999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.244 × 10⁹⁵(96-digit number)

12442841353589479020…18755740110203760001

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

2.488 × 10⁹⁵(96-digit number)

24885682707178958041…37511480220407519999

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