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Block #2,045,651

TWNLength 10β˜…β˜…β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 3/30/2017, 8:55:23 PM Β· Difficulty 10.6822 Β· 4,796,445 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
86d1e4703878884f13c2c62cf46d19a847f6e56459ef1bbf8f89a50ae66ad0ca
View on Chainz β†—

Height

#2,045,651

Difficulty

10.682185

Transactions

1

Size

198 B

Version

2

Bits

0aaea3ad

Nonce

702,765,981

Timestamp

3/30/2017, 8:55:23 PM

Confirmations

4,796,445

Mined by

⛏️ P2SHAJa5jbYR34Epx8DQCU771aMb7mn8jqH6XP

Merkle Root

9c309d9f1674616d6b8224762809b4cd3d52d0e33190482b482d5fcc97fa1ddf
Transactions (1)
Coinbase9c309d9f167461…2d5fcc97fa1ddf
1 in β†’ 1 out8.7500 XPM109 B
AJa5jbYR…n8jqH6XP
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.

2.657 Γ— 10⁹³(94-digit number)
26574346414179644863…75856802918529592639
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
2.657 Γ— 10⁹³(94-digit number)
26574346414179644863…75856802918529592639
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
2.657 Γ— 10⁹³(94-digit number)
26574346414179644863…75856802918529592641
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
5.314 Γ— 10⁹³(94-digit number)
53148692828359289726…51713605837059185279
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
5.314 Γ— 10⁹³(94-digit number)
53148692828359289726…51713605837059185281
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
1.062 Γ— 10⁹⁴(95-digit number)
10629738565671857945…03427211674118370559
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
1.062 Γ— 10⁹⁴(95-digit number)
10629738565671857945…03427211674118370561
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
2.125 Γ— 10⁹⁴(95-digit number)
21259477131343715890…06854423348236741119
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
2.125 Γ— 10⁹⁴(95-digit number)
21259477131343715890…06854423348236741121
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
4.251 Γ— 10⁹⁴(95-digit number)
42518954262687431781…13708846696473482239
Verify on FactorDB β†—Wolfram Alpha β†—
2^4 Γ— origin + 1
4.251 Γ— 10⁹⁴(95-digit number)
42518954262687431781…13708846696473482241
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^4 Γ— origin + 1 βˆ’ 2^4 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)

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 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
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 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:57,981,154 XPMΒ·at block #6,842,095 Β· updates every 60s
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