Rabbit Hole #2 - Consecutive *Odd* Semi-Primes

Audience: Number Nerds

Moderate Jargon Alert: This post is primarily aimed at people who are comfortable using numbers as play objects, so the post’s jargon rating is moderate. New to this? don’t let that put you off, we all started somewhere!

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I know what you’re thinking: “Why Vanessa, that’s an odd question”, to which I’d respond, “Well, it’s not as fun when it’s even” * dodges tomatoes *.

Bad jokes aside, the keenest readers may recall a previous rabbit hole addressing strings of consecutive semi-primes (see Rabbit Hole #1). Well, asking about consecutive odd semi-primes was the natural next step… obviously…

Plain Language Question: Can you find a pair of consecutive odd numbers (odd numbers that differ from each other by two, e.g. 1 and 3) that are both semi-prime (have precisely two, not necessarily distinct, prime factors, e.g. \(4=2\times 2\) or \(15 = 3\times 5\))? Can you find a triple? A quadruple? What is the longest possible chain of consecutive odd semi-primes? Is there one?

And a mathematical language version for my colleagues or those otherwise inclined:

Question (Mathematically expressed): Can you find \(n\in\mathbb{Z}_{\geq 0}\) such that all the terms in the sequence \(a_k(n) = 2(n+k)+1\) are semi-prime, where \(k \in \{1,\dots,K\}, K\in\mathbb{Z}_{\geq 1}\)? Does \(n\) exist for all \(K\)?

Readers who are less familiar with mathematical terminology may find it interesting to try to translate the mathematical symbols using the plain English version provided. Isn’t language learning fun?!

Note: For all of the sequence lengths that follow, the sequence I’ve included is the first one (with that length) of consecutive odd semi-primes you’ll encounter when counting up from zero.

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Pairs, Triples and Quadruples (\(k=1,2,3\))

Do you remember that code I wrote for the quadruple of consecutive semi-primes?¹ Yes, the code I wrote without pausing to think that ended up being completely redundant and a bit of a facepalm? WELL ITS PRETTY DAMN USEFUL NOW! Though, on a matter of principle and mathematician’s pride, I did do the first pair of consecutive odd semi-primes by hand:

Sequence Length	Sequence Terms *
2	33, 35
3	91, 93, 95
4	299, 301, 303, 305

* Verifying that the sequence terms are indeed semi-prime is a (hazing) exercise left for the reader.

An immediately apparent pattern is that as the number of terms in the sequence increases, the size of the numbers in the sequence also increases.

Then something interesting happened: I looked at a sequence with five terms:

Quintuples of Consecutive Odd Semi-Primes (\(k=4\))

Sequence Length	Sequence Terms
4	299, 301, 303, 305
5	213, 215, 217, 219, 221

(these are the first quads and quints)

The numbers in the quintuple are smaller!

It is tempting to make a statement like “as the length of the sequence increases, the size of the numbers in the sequence also increases”. This statement ‘feels’ like that should be true, and it matches our precedent, but this is a lovely example of why we need to prove things. While the feelings we get about numbers should be noted, they should not be given ultimate rule; in this neighbourhood, a feeling is not enough.

In terms of why the values in the quintuple are smaller, you might find this annoying, but I think it just happened to end up that way (but I’d be delighted to be wrong about that). I wonder if the random distribution of the primes among whole numbers has interesting consequences on the distribution of semi-primes, but that’s for another day.

Sextuples and more (\(k \geq 5\))

Given what we learned about consecutive semi-primes and how there can’t be more than 3 of them ¹, I went into this with suspicions that the max number of consecutive odd semi-primes would be (at most) 8 (see if you can figure out why while I show you the next few sequences):

Sequence Length	Sequence Terms
6	1383, 1385, 1387, 1389, 1391, 1393
7	3091, 3093, 3095, 3097, 3099, 3101, 3103
8	8129, 8131, 8133, 8135, 8137, 8139, 8141, 8143

So, what do you think? Can we go higher?

No.

Since we are dealing with odd numbers, there are no multiples of four to worry about as in Rabbit Hole #1. The next potential spanner in the semi-prime sequence works is the next square—nine.

In nine (or more) consecutive odd numbers, we will bump into a multiple of nine, and if we bump into a multiple of nine, then that number would be \(3\times 3 \times something else\), and that is not semi-prime!

So, in answer to the question

What is the longest possible chain of consecutive odd semi-primes? Is there one?

There is indeed a longest possible chain of consecutive odd semi-primes, and it has a length of eight.

Other Rabbit Holes to Explore

I have been resisting the mathematician’s urge to abstract, but if you’ve stuck with me this far, I doubt you’ll mind me indulging in a bit of abstraction.

The chief question for me here is, does this continue? What about the longest sequence of semi-primes with a difference of three or four? Are there surprising/lovely/curious patterns in semi-prime sequences with non-fixed deterministic spacings, e.g. spacing increasing by one for each successive term in the sequence? Will the presence of a square in these more abstract sequences be unavoidable, ensuring a finite longest semi-prime sequence for these other spacings?

I don’t know, but what do you think?

1. Reference to Rabbit Hole #1 ↩︎ ↩︎²