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Block #167,109

TWNLength 9β˜…β˜†β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 9/16/2013, 9:00:17 AM Β· Difficulty 9.8688 Β· 6,643,180 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
847634b2631ad1f32a6fbbd4d15baaeada3f1e07d7a2c8141bfd5fb1f385f5fb
View on Chainz β†—

Height

#167,109

Difficulty

9.868764

Transactions

2

Size

391 B

Version

2

Bits

09de6759

Nonce

24,180

Timestamp

9/16/2013, 9:00:17 AM

Confirmations

6,643,180

Mined by

⛏️ P2SHAbre2hUpbdZBPFEEyixCfvGQHmim3QyaUZ

Merkle Root

f1fa02bceb2cd7415e2bbe7dc0e9fef296055825705ab30e5cd2d0e0e9392a7e
Transactions (2)
Coinbasedc9254b1226ce9…53c99979695542
1 in β†’ 1 out10.2600 XPM109 B
Abre2hUp…im3QyaUZ
9f69b2d113f396…2ea7cacc447353
1 in β†’ 2 out10.2600 XPM192 B
ARME3NdA…EQzCGHC9AFz9fXym…wh4UqpkL
Prime Chain Origin

This is the prime chain origin stored in the block header. It is a composite number (not prime itself) β€” it equals the first prime in the chain multiplied by a primorial. The origin anchors the entire chain to this specific block.

5.048 Γ— 10⁹³(94-digit number)
50486826962630090673…05169163926067343359
Discovered Prime Numbers
Lower: 2^k Γ— origin βˆ’ 1 | Upper: 2^k Γ— 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 β€” Twin Prime Pair (origin Β± 1)
origin βˆ’ 1
5.048 Γ— 10⁹³(94-digit number)
50486826962630090673…05169163926067343359
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
5.048 Γ— 10⁹³(94-digit number)
50486826962630090673…05169163926067343361
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: origin + 1 βˆ’ origin βˆ’ 1 = 2 (twin primes βœ“)
Level 1 β€” Twin Prime Pair (2^1 Γ— origin Β± 1)
2^1 Γ— origin βˆ’ 1
1.009 Γ— 10⁹⁴(95-digit number)
10097365392526018134…10338327852134686719
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
1.009 Γ— 10⁹⁴(95-digit number)
10097365392526018134…10338327852134686721
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^1 Γ— origin + 1 βˆ’ 2^1 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 2 β€” Twin Prime Pair (2^2 Γ— origin Β± 1)
2^2 Γ— origin βˆ’ 1
2.019 Γ— 10⁹⁴(95-digit number)
20194730785052036269…20676655704269373439
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
2.019 Γ— 10⁹⁴(95-digit number)
20194730785052036269…20676655704269373441
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^2 Γ— origin + 1 βˆ’ 2^2 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 3 β€” Twin Prime Pair (2^3 Γ— origin Β± 1)
2^3 Γ— origin βˆ’ 1
4.038 Γ— 10⁹⁴(95-digit number)
40389461570104072538…41353311408538746879
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
4.038 Γ— 10⁹⁴(95-digit number)
40389461570104072538…41353311408538746881
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^3 Γ— origin + 1 βˆ’ 2^3 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 4 β€” Twin Prime Pair (2^4 Γ— origin Β± 1)
2^4 Γ— origin βˆ’ 1
8.077 Γ— 10⁹⁴(95-digit number)
80778923140208145077…82706622817077493759
Verify on FactorDB β†—Wolfram Alpha β†—

What this block proved

The miner who found this block proved the existence of 9 consecutive prime numbers forming a Bi-Twin Chain. The prime chain origin β€” the large number shown above β€” anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

β˜…β˜†β˜†β˜†β˜†
Rarity
CommonChain length 9

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 Γ— 3 Γ— 5 Γ— 7 Γ— …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial β€” that divisibility is part of the proof.

Prime Chain Origin = First Prime Γ— Primorial (2Β·3Β·5Β·7Β·11·…)
Source: Primecoin prime.cpp β€” CheckPrimeProofOfWork()

This is why the origin has many small prime factors β€” 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 βˆ’ 1 and p+2 = origin/primorial + 1
Cross-reference on Chainz Explorer
Circulating Supply:57,726,387 XPMΒ·at block #6,810,288 Β· updates every 60s
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AQZeJoQ81VgeHQBpEYTgvNqV1gnz3L5SvK

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