Block #298,578

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/7/2013, 10:25:48 AM · Difficulty 9.9920 · 6,512,577 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

ba756adc8b7750b72e1ebfa4ae20da0581369e28eb5edfab3ae13ffc5c94736a

View on Chainz ↗

Height

#298,578

Difficulty

9.991956

Transactions

Size

1.77 KB

Version

Bits

09fdf0cf

Nonce

429,128

Timestamp

12/7/2013, 10:25:48 AM

Confirmations

6,512,577

Mined by

⛏️ RaRAZ2pPuKgN7GXPJPBLxtm9bkWcE95SN2Yr7

Merkle Root

803eb580dd35f3a486400e794c2a2ac9a88bb35b22fb1375542ad931d6ac529c

Transactions (4)

Coinbase72bba409812f99…243689754c2671

1 in → 29 out9.9800 XPM1.02 KB

AZ2pPuKg…95SN2Yr7 AKEwr54i…9BfmMkRh ATr7rJvZ…v4W3ybba AWA5FEzr…x9RdPKTy AS56dfmh…9PNcZwS6

42c9a434f3335d…6c2a08ff01204f

1 in → 2 out398.0110 XPM225 B

AUwBCP8o…o4HgeVME AZDUmrFV…nPRRhaM7

32aeb229866a5d…d7bbe08c8935d5

1 in → 2 out395.7894 XPM225 B

AaVFHFSq…bCnX1Bpw AUiPraHa…JFpsb5Fq

ef9b306c715f6e…8826d41597391a

1 in → 2 out393.5963 XPM226 B

ARo7tFxX…4fNfiVwj AKUEaUvA…6nAqWbP5

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.

2.888 × 10⁹³(94-digit number)

28887116885737612308…03347027706010107999

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

2.888 × 10⁹³(94-digit number)

28887116885737612308…03347027706010107999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.888 × 10⁹³(94-digit number)

28887116885737612308…03347027706010108001

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

5.777 × 10⁹³(94-digit number)

57774233771475224616…06694055412020215999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.777 × 10⁹³(94-digit number)

57774233771475224616…06694055412020216001

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

1.155 × 10⁹⁴(95-digit number)

11554846754295044923…13388110824040431999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.155 × 10⁹⁴(95-digit number)

11554846754295044923…13388110824040432001

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

2.310 × 10⁹⁴(95-digit number)

23109693508590089846…26776221648080863999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.310 × 10⁹⁴(95-digit number)

23109693508590089846…26776221648080864001

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

4.621 × 10⁹⁴(95-digit number)

46219387017180179693…53552443296161727999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.621 × 10⁹⁴(95-digit number)

46219387017180179693…53552443296161728001

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 #298,578

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