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Block #164,245

TWNLength 9β˜…β˜†β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 9/14/2013, 1:33:13 PM Β· Difficulty 9.8618 Β· 6,644,155 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
c692f84ce94e3c877c851914b9eaa7ea4133c548e39173b8a16b527a04541a1b
View on Chainz β†—

Height

#164,245

Difficulty

9.861796

Transactions

2

Size

6.55 KB

Version

2

Bits

09dc9eb2

Nonce

54,116

Timestamp

9/14/2013, 1:33:13 PM

Confirmations

6,644,155

Mined by

⛏️ P2SHASbwzPTTJ4Dogjio1ragQ3EydoXC3H7o8k

Merkle Root

783f8934b27df74c9550278bd4d69e7e6a717d6e5f1c9a663ac87dc032ba74f0
Transactions (2)
Coinbasedb59565d81a907…2eae13d3fc1a88
1 in β†’ 1 out10.3463 XPM109 B
ASbwzPTT…XC3H7o8k
43e144df796583…af916808b194d8
46 in β†’ 1 out307.9400 XPM6.36 KB
AV6URp9C…SYeCgYBK
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.

4.023 Γ— 10⁹³(94-digit number)
40239122700688111255…88416162398328463399
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
4.023 Γ— 10⁹³(94-digit number)
40239122700688111255…88416162398328463399
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
4.023 Γ— 10⁹³(94-digit number)
40239122700688111255…88416162398328463401
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
8.047 Γ— 10⁹³(94-digit number)
80478245401376222511…76832324796656926799
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
8.047 Γ— 10⁹³(94-digit number)
80478245401376222511…76832324796656926801
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
1.609 Γ— 10⁹⁴(95-digit number)
16095649080275244502…53664649593313853599
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
1.609 Γ— 10⁹⁴(95-digit number)
16095649080275244502…53664649593313853601
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
3.219 Γ— 10⁹⁴(95-digit number)
32191298160550489004…07329299186627707199
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
3.219 Γ— 10⁹⁴(95-digit number)
32191298160550489004…07329299186627707201
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
6.438 Γ— 10⁹⁴(95-digit number)
64382596321100978009…14658598373255414399
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:57,711,257 XPMΒ·at block #6,808,399 Β· updates every 60s
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AQZeJoQ81VgeHQBpEYTgvNqV1gnz3L5SvK

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