Block #31,217

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/13/2013, 10:32:04 PM · Difficulty 7.9883 · 6,779,235 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

7b52468b7c6549405ffe258804ad1978b225a239019e614a70306e6e35fd96f0

View on Chainz ↗

Height

#31,217

Difficulty

7.988303

Transactions

Size

197 B

Version

Bits

07fd0174

Nonce

Timestamp

7/13/2013, 10:32:04 PM

Confirmations

6,779,235

Mined by

⛏️ P2SHAKSwr9eUsXtYAeGKZ6MmF96jvsdLNxuVyn

Merkle Root

e33d6274efa0bebb246c4780cde190883cba270a80e4ab5f1ccaa45b502b49eb

Transactions (1)

Coinbasee33d6274efa0be…caa45b502b49eb

1 in → 1 out15.6500 XPM108 B

AKSwr9eU…dLNxuVyn

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.

7.929 × 10⁹²(93-digit number)

79292849079862204489…06925399886352386619

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

7.929 × 10⁹²(93-digit number)

79292849079862204489…06925399886352386619

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.929 × 10⁹²(93-digit number)

79292849079862204489…06925399886352386621

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

1.585 × 10⁹³(94-digit number)

15858569815972440897…13850799772704773239

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.585 × 10⁹³(94-digit number)

15858569815972440897…13850799772704773241

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

3.171 × 10⁹³(94-digit number)

31717139631944881795…27701599545409546479

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.171 × 10⁹³(94-digit number)

31717139631944881795…27701599545409546481

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

6.343 × 10⁹³(94-digit number)

63434279263889763591…55403199090819092959

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.343 × 10⁹³(94-digit number)

63434279263889763591…55403199090819092961

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

Block #31,217

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