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Block #40,686

TWNLength 8β˜…β˜†β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 7/14/2013, 3:24:40 PM Β· Difficulty 8.4561 Β· 6,768,850 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
5d0b47f4dc72a0aa29f4dfbc06c54ee47424bc2cb7c607ccf742dce0a289d55b
View on Chainz β†—

Height

#40,686

Difficulty

8.456118

Transactions

2

Size

357 B

Version

2

Bits

0874c428

Nonce

646

Timestamp

7/14/2013, 3:24:40 PM

Confirmations

6,768,850

Mined by

⛏️ P2SHAGH5eFN1WveRyeeudfgVwqXTStv1mzD1xx

Merkle Root

bd6df91d4888cd6ab404a09b40df327f2fc6a5ed5ffefe70ce807f407d2fa4e8
Transactions (2)
Coinbaseb77ec80d2a0ea5…a6b8a9bde124e4
1 in β†’ 1 out13.9800 XPM109 B
AGH5eFN1…v1mzD1xx
2593673f223564…46e221c9cb2c21
1 in β†’ 1 out15.4800 XPM157 B
AXUtFnVB…2K4kmJnq
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.

5.670 Γ— 10⁹⁷(98-digit number)
56707347762585115932…31230411573103594309
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
5.670 Γ— 10⁹⁷(98-digit number)
56707347762585115932…31230411573103594309
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
5.670 Γ— 10⁹⁷(98-digit number)
56707347762585115932…31230411573103594311
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
1.134 Γ— 10⁹⁸(99-digit number)
11341469552517023186…62460823146207188619
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
1.134 Γ— 10⁹⁸(99-digit number)
11341469552517023186…62460823146207188621
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
2.268 Γ— 10⁹⁸(99-digit number)
22682939105034046373…24921646292414377239
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
2.268 Γ— 10⁹⁸(99-digit number)
22682939105034046373…24921646292414377241
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
4.536 Γ— 10⁹⁸(99-digit number)
45365878210068092746…49843292584828754479
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
4.536 Γ— 10⁹⁸(99-digit number)
45365878210068092746…49843292584828754481
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^3 Γ— origin + 1 βˆ’ 2^3 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)

What this block proved

The miner who found this block proved the existence of 8 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 8

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:57,720,367 XPMΒ·at block #6,809,535 Β· updates every 60s
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AQZeJoQ81VgeHQBpEYTgvNqV1gnz3L5SvK

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