Block #311,829

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 6:10:35 PM · Difficulty 9.9956 · 6,494,987 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

047e0fc10de2a76ef306c4446a13af2ebe8c3e689a601df63afc27fe99585eb2

View on Chainz ↗

Height

#311,829

Difficulty

9.995607

Transactions

Size

1.61 KB

Version

Bits

09fee016

Nonce

466,372

Timestamp

12/14/2013, 6:10:35 PM

Confirmations

6,494,987

Mined by

⛏️ aRAQwd2tMyW8G9GsusvJvhMbrCCBCzytqW6x

Merkle Root

95e85e9b9da2cbeeb075d9b91299e6d21b39935166d115797ea86413856b0556

Transactions (4)

Coinbasef9132861a1b9ea…d1effa6f2f24b5

1 in → 24 out9.9900 XPM879 B

AQwd2tMy…CzytqW6x AcuM7XiJ…5uND8Jce AQwRN8bE…V8PsTcY9 Aac9Hjdg…P4ktypc3 AG5YAjec…VhA4Aeh1

fb712273899f8c…a0604ff80702d8

1 in → 2 out5.6400 XPM226 B

AXXnPyL8…nRGoG6DX ALPNRxL5…Q93ycKpR

fb1582279d5a37…f88359b3517087

1 in → 2 out4.5513 XPM225 B

Ad5BAt58…7MwkrfrM AKWqoZJG…Pqsk51BZ

b3472dddd91ae7…08c7330558c1f6

1 in → 2 out3.6233 XPM225 B

AFutdTaZ…VSUfHUeJ AZNURcWM…EA4AMHAh

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.

2.523 × 10⁹³(94-digit number)

25232171771752180384…38649794101865977599

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

2.523 × 10⁹³(94-digit number)

25232171771752180384…38649794101865977599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.523 × 10⁹³(94-digit number)

25232171771752180384…38649794101865977601

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

5.046 × 10⁹³(94-digit number)

50464343543504360769…77299588203731955199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.046 × 10⁹³(94-digit number)

50464343543504360769…77299588203731955201

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

1.009 × 10⁹⁴(95-digit number)

10092868708700872153…54599176407463910399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.009 × 10⁹⁴(95-digit number)

10092868708700872153…54599176407463910401

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

2.018 × 10⁹⁴(95-digit number)

20185737417401744307…09198352814927820799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.018 × 10⁹⁴(95-digit number)

20185737417401744307…09198352814927820801

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

4.037 × 10⁹⁴(95-digit number)

40371474834803488615…18396705629855641599

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.037 × 10⁹⁴(95-digit number)

40371474834803488615…18396705629855641601

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 #311,829

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