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Say we have a function (loosely speaking) which takes an input of an integer x1x2x3x4..xn, where xn is the nth digit in the integer. This function sums the digits of this number modulo(9) and maps it to this number.

So f(19)-> (1+9)modulo 9 -> 1 , f(104729) -> (1+0+4+7+2+9)modulo 9 -> 5. f(209459) -> 2.

Edit: I just realised that: g(x) = x(modulo 9), ie g(19)-> 19 (modulo 9) -> 1 etc gives the same result. Much simpler.

I've noticed that applying this 'function' to the primes in many Cunningham chains (could it be all?) gives a distinct pattern.

All primes in these chains will either map to a sequence of 2's and 5's, starting with either 2 or 5 or all 8's.

https://primefan.tripod.com/CunninghamChains.html

Gives a list of chains and you can apply this function to each and see the pattern.

eg: (104369, 208739, 417479, 834959)->(5,2,5,2)

(43541,87083) -> (8,8)

I guess my overall question is, does there exist a chain that breaks this pattern and if not then could this could help in finding more chains?

user2814916
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1 Answers1

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Some observations. Recall the definition of a Cunningham chain (of the first kind), it is a sequence of prime numbers $(p_1, \ldots, p_n)$ such that $p_{i + 1} = 2p_{i} + 1$.

Looking modulo $9$, you thus have that $p_{i + 1} \equiv 2p_i + 1 \mod 9$. Thus the value of $p_1$ modulo $9$ determines all subsequent values. In particular, this means that for a chain $(p_1, \ldots, p_n)$, the values modulo $9$ follow one of three cycles;

$$\ldots \to 0 \to 1 \to 3 \to 7 \to 6 \to 4 \to 0 \ldots,$$ $$\ldots \to 2 \to 5 \to 2 \to 5 \to \ldots,$$ or $$\ldots \to 8 \to 8 \to \ldots$$

Now, the chains that break this pattern are precisely those for which $p_1 \mod 9$ is not equivalent to $2, 5$ or $8$. However, by looking at the cycle such a chain falls in, these can be of at most length $3$, since if the chain is longer then we are guaranteed to hit $p_i \equiv 3$ or $p_i \equiv 6 \mod 9$.

In either case, $p_i$ is divisible by $3$ hence not prime. The only exception is the case $p_1 = 3$, which gives a chain of length $2$.

For finding chains, the only real use this has is being able to tell whether a prime has a chance of starting a cycle of length longer than $3$.

Rushy
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