质币和卡宁汉素数链

质币和卡宁汉素数链
Primecoin and Cunningham Prime Chains

原始链接: https://www.johndcook.com/blog/2026/01/10/prime-chains/

这篇帖子探讨了“链”在两个领域中的应用：数论和加密货币。在数论中，它关注**坎宁安链**，即一系列素数，每个数几乎是前一个数的两倍（2p ± 1）。链可以是第一类（2p+1）或第二类（2p-1）。虽然目前尚不清楚是否存在无限多个，但数学家推测存在所有长度的无限链——已知最长的分别是第一类和第二类，长度分别为17和19。文中还提到了**双生链**，它连接了形成两种类型的坎宁安链的素数。帖子随后转向**Primecoin**，一种早期加密货币，利用素链发现作为其“工作量证明”。矿工通过找到指定长度的素链（坎宁安链或双生链），并将其与区块的哈希值关联来赚取Primecoin。难度通过改变所需的链长度进行调整，从而激励持续的素数探索。

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原文

The title of this post has a double meaning. We will look at chains in the sense of number theory and in the sense of cryptocurrency, i.e. Cunningham chains and blockchains, that involve prime numbers.

Cunningham chains

A chain of primes is a sequence of prime numbers in which each is almost double its predecessor. That is, the next number after p is 2p ± 1.

In a Cunningham chain of the first kind, the successor of p is 2p + 1. For example,

41, 83, 167.

In a Cunningham chain of the second kind, the successor of p is 2p − 1. For example,

19, 37, 73.

Two questions come up immediately. First, are there infinitely many Cunningham chains? Second, how long can a Cunningham chain be? What is known and what is conjectured are at opposite ends of the spectrum. It is unknown whether there are infinitely many Cunningham chains of length 2, but it is conjectured that there are infinitely many Cunningham chains of all lengths.

According to this page, the longest known Cunningham chains of the first kind has length 17, and the longest known Cunningham chain of the second kind has length 19. We can verify these results with the following Python code.

from sympy import isprime

def chain_length(start, kind):
    p = start
    c = 0
    while isprime(p):
        c += 1
        p = 2*p + kind
    return c

print(chain_length(2759832934171386593519, 1))
print(chain_length(79910197721667870187016101, -1))

Bi-twin chains

A number n is the basis of a bi-twin chain of length k if n − 1 is the start of a Cunningham chain of the first kind of length k and n + 1 is the start of a Cunningham chain of the second kind of length k.

I say more about bi-twin prime chains in the next post.

Primecoin

Primecoin was one of the first cryptocurrencies, coming out four years after Bitcoin. Primecoin is still going, though its market cap is six orders magnitude smaller than that of Bitcoin.

What’s interesting about Primecoin is that it uses finding prime chains as its proof of work task [1]. To mint a new Primecoin block, you must find a prime chain of the required length whose origin a multiple of the hash of the block header [2].

Primecoin allows any of the three kinds of prime chains mentioned above: Cunningham chains of the first or second kind, or a bi-twin prime chain. Primecoin adjusts its mining difficulty over time by varying the length of Cunningham or bi-twin chain needed to mint a new block.

[1] Strictly speaking, Primecoin requires finding probable prime chains, as explained here.

[2] The origin of a prime chain is n if the first item in a Cunningham chain of the first kind is is n + 1, or if the first item in a Cunningham chain of the first kind or a bi-twin chain is n − 1.