Block #698,323

TWNLength 11★★★☆☆

Bi-Twin Chain · Discovered 8/29/2014, 12:05:15 PM · Difficulty 10.9591 · 6,127,979 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b003a7f49e7ce47f46816dc0b01afefff1db3223d350477a1d7b41660fd170c6

View on Chainz ↗

Height

#698,323

Difficulty

10.959075

Transactions

Size

796 B

Version

Bits

0af585f8

Nonce

308,312,205

Timestamp

8/29/2014, 12:05:15 PM

Confirmations

6,127,979

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

0047e65eea813b0c95b8889b04062cadcffc38bb195e2e9d0ee506a29e3843aa

Transactions (2)

Coinbase81930dbd406058…93a2d53b76ec11

1 in → 1 out8.3200 XPM116 B

ANSXnLY2…Sd25TjkD

b28e9905e0adf5…e5e0ec0aa20bb3

3 in → 4 out7.4495 XPM589 B

AGqSWjWQ…gzSSkRFQ AHXJ1dhk…mhL46N6m AVuR57uD…qdXNw3x1 AeKNrjNj…1NEq95Ta

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.

2.058 × 10⁹⁶(97-digit number)

20587613405190531017…64476643554016682959

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

2.058 × 10⁹⁶(97-digit number)

20587613405190531017…64476643554016682959

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.058 × 10⁹⁶(97-digit number)

20587613405190531017…64476643554016682961

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

4.117 × 10⁹⁶(97-digit number)

41175226810381062035…28953287108033365919

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.117 × 10⁹⁶(97-digit number)

41175226810381062035…28953287108033365921

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

8.235 × 10⁹⁶(97-digit number)

82350453620762124070…57906574216066731839

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.235 × 10⁹⁶(97-digit number)

82350453620762124070…57906574216066731841

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

1.647 × 10⁹⁷(98-digit number)

16470090724152424814…15813148432133463679

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.647 × 10⁹⁷(98-digit number)

16470090724152424814…15813148432133463681

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

3.294 × 10⁹⁷(98-digit number)

32940181448304849628…31626296864266927359

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.294 × 10⁹⁷(98-digit number)

32940181448304849628…31626296864266927361

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

6.588 × 10⁹⁷(98-digit number)

65880362896609699256…63252593728533854719

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

Block #698,323

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