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Block #413,514

TWNLength 10β˜…β˜…β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 2/21/2014, 7:18:55 AM Β· Difficulty 10.4145 Β· 6,403,714 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
de7dd18d9bc070ba4eca6a1e059ee05ce8a6def953d8f917ac54591ded7c8309
View on Chainz β†—

Height

#413,514

Difficulty

10.414505

Transactions

1

Size

200 B

Version

2

Bits

0a6a1d01

Nonce

123,967

Timestamp

2/21/2014, 7:18:55 AM

Confirmations

6,403,714

Merkle Root

e146e3de88a1598c5b2f9cdbbe079a14358c88e64fb5bfa49f43b3240a991e6f
Transactions (1)
Coinbasee146e3de88a159…43b3240a991e6f
1 in β†’ 1 out9.2100 XPM110 B
AeKNrjNj…1NEq95Ta
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.

7.862 Γ— 10⁹⁡(96-digit number)
78627700547586062133…72673087799669489920
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
7.862 Γ— 10⁹⁡(96-digit number)
78627700547586062133…72673087799669489919
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
7.862 Γ— 10⁹⁡(96-digit number)
78627700547586062133…72673087799669489921
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
1.572 Γ— 10⁹⁢(97-digit number)
15725540109517212426…45346175599338979839
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
1.572 Γ— 10⁹⁢(97-digit number)
15725540109517212426…45346175599338979841
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
3.145 Γ— 10⁹⁢(97-digit number)
31451080219034424853…90692351198677959679
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
3.145 Γ— 10⁹⁢(97-digit number)
31451080219034424853…90692351198677959681
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
6.290 Γ— 10⁹⁢(97-digit number)
62902160438068849706…81384702397355919359
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
6.290 Γ— 10⁹⁢(97-digit number)
62902160438068849706…81384702397355919361
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
1.258 Γ— 10⁹⁷(98-digit number)
12580432087613769941…62769404794711838719
Verify on FactorDB β†—Wolfram Alpha β†—
2^4 Γ— origin + 1
1.258 Γ— 10⁹⁷(98-digit number)
12580432087613769941…62769404794711838721
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.

View full Prime Chain Discovery page β†’
β˜…β˜…β˜†β˜†β˜†
Rarity
UncommonChain length 10
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
Verify on a Primecoin Node

Anyone running a Primecoin node can independently verify this block using the following RPC commands. Run these from the primecoin-cli command line or via the debug console in the Primecoin wallet.

1. Get block hash by height
getblockhash 413514

Returns the block hash for a given block height. Use this to confirm the hash shown above matches the chain.

2. Get full block data
getblock de7dd18d9bc070ba4eca6a1e059ee05ce8a6def953d8f917ac54591ded7c8309

Returns the full block header including difficulty, prime chain origin, prime chain type, transaction IDs, and all other fields shown on this page.

How to run these commands: To verify this data independently, open a terminal on your own Primecoin node and run primecoin-cli <command>. Alternatively, open the Primecoin-Qt wallet, go to Help β†’ Debug Window β†’ Console, and type the command directly. The node must be fully synced to this block height for the commands to return results.

Cross-reference on Chainz Explorer

Chainz is an independent Primecoin block explorer. Compare this block's data to verify accuracy.

View Block #413,514 on Chainz β†—
Circulating Supply:57,781,862 XPMΒ·at block #6,817,227 Β· updates every 60s
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