Block #311,070

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 9:23:26 AM · Difficulty 9.9954 · 6,499,478 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b21112942e75fff33f2f3e69383f9e27215c4445f0344cc02e5e68990ec3adfc

View on Chainz ↗

Height

#311,070

Difficulty

9.995376

Transactions

Size

863 B

Version

Bits

09fed0fd

Nonce

91,114

Timestamp

12/14/2013, 9:23:26 AM

Confirmations

6,499,478

Mined by

⛏️ aR"bAKxxLGqhSp8N6q1niks6PPt79QoySJc6b5

Merkle Root

7b74e9b785c518747b19b4a5b85af221fbdb0382eb27e074f9aa3a31bc0efa06

Transactions (4)

Coinbase3c65857b917599…007db8917c471d

1 in → 1 out9.9900 XPM97 B

AKxxLGqh…oySJc6b5

b8e6787f0adda4…7e3721dd15cdbb

1 in → 2 out7.8785 XPM226 B

APjvnWAy…KnKfTuoA ASPSQx9K…P4tK74ET

33e20296bccffd…3206c65d830d92

1 in → 2 out1.4473 XPM225 B

AbWuaqd8…zxTzepN5 Ac9Lzw78…JA877Fwb

e5bdab9e5a9a19…e8564c4d03e1ce

1 in → 2 out1.7073 XPM226 B

AaEhVQn8…moggStG3 AGUiHrXX…99nzH74G

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.

5.074 × 10⁹¹(92-digit number)

50742515450146371481…64109845099111931749

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

5.074 × 10⁹¹(92-digit number)

50742515450146371481…64109845099111931749

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.074 × 10⁹¹(92-digit number)

50742515450146371481…64109845099111931751

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

1.014 × 10⁹²(93-digit number)

10148503090029274296…28219690198223863499

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.014 × 10⁹²(93-digit number)

10148503090029274296…28219690198223863501

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

2.029 × 10⁹²(93-digit number)

20297006180058548592…56439380396447726999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.029 × 10⁹²(93-digit number)

20297006180058548592…56439380396447727001

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

4.059 × 10⁹²(93-digit number)

40594012360117097185…12878760792895453999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.059 × 10⁹²(93-digit number)

40594012360117097185…12878760792895454001

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

8.118 × 10⁹²(93-digit number)

81188024720234194370…25757521585790907999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.118 × 10⁹²(93-digit number)

81188024720234194370…25757521585790908001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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 #311,070

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