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Block #166,721

TWNLength 9β˜…β˜†β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 9/16/2013, 2:40:22 AM Β· Difficulty 9.8685 Β· 6,628,012 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
27239bc7c3e6c126fb85c3ff6c92cd9b02d34dcaf41f81348417b3c943b8d1b4
View on Chainz β†—

Height

#166,721

Difficulty

9.868508

Transactions

1

Size

199 B

Version

2

Bits

09de5684

Nonce

25,426

Timestamp

9/16/2013, 2:40:22 AM

Confirmations

6,628,012

Mined by

⛏️ P2SHAKhaGKFX4s17sd56QzVFtsa8mSKjeAXtBv

Merkle Root

71dfff4b2b48809c808e213641aad69663cb26a0829579789e187c8e8b3f7794
Transactions (1)
Coinbase71dfff4b2b4880…187c8e8b3f7794
1 in β†’ 1 out10.2500 XPM109 B
AKhaGKFX…KjeAXtBv
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.

9.175 Γ— 10⁹⁴(95-digit number)
91759359852358860314…88061196990746567999
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
9.175 Γ— 10⁹⁴(95-digit number)
91759359852358860314…88061196990746567999
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
9.175 Γ— 10⁹⁴(95-digit number)
91759359852358860314…88061196990746568001
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
1.835 Γ— 10⁹⁡(96-digit number)
18351871970471772062…76122393981493135999
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
1.835 Γ— 10⁹⁡(96-digit number)
18351871970471772062…76122393981493136001
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
3.670 Γ— 10⁹⁡(96-digit number)
36703743940943544125…52244787962986271999
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
3.670 Γ— 10⁹⁡(96-digit number)
36703743940943544125…52244787962986272001
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
7.340 Γ— 10⁹⁡(96-digit number)
73407487881887088251…04489575925972543999
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
7.340 Γ— 10⁹⁡(96-digit number)
73407487881887088251…04489575925972544001
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
1.468 Γ— 10⁹⁢(97-digit number)
14681497576377417650…08979151851945087999
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:57,601,915 XPMΒ·at block #6,794,732 Β· updates every 60s
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