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Are there infinitely many primes none of whose proper initial segments are primes? An example of those primes is $88547$, because $8,88,885$ and $8854$ are not primes. The first few terms in the sequence (A069090) are:

   2, 3, 5, 7, 11, 13, 17, 19, 41, 43, 47, 61, 67, 83, 89, 97, 101, 103, 107, 109, 127, 149, 151, 157, 163, 167, 181, 401, 409, 421, 443, 449, 457, 461, 463, 467, 487, 491, 499, 601, 607, 631, 641, 643, 647, 653, 659, 661, 683, 691, 809, 811, 821, 823, 827

Note: this question has been raised in the comments of a similar question about the convergence of the sum of the reciprocals of A069090, so this could be relevant.

François Huppé
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1 Answers1

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To prove this, we can use another closely related sequence: composites none of whose proper initial segments are primes. In order to lighten the text, let's define the following references:

Type $α$ integers: composites none of whose proper initial segments are primes.

Type $β$ integers: primes none of whose proper initial segments are primes.

Fact $1$: It is trivially true that there are infinitely many type $α$ integers, because any sequence of even digits of any length that does not start with $2$ is a type $α$ integer.

Fact $2$: Any prime that is not a type $β$ integer, by definition, have $1$ or more prime initial segments, and the smallest of those segments is a type $β$ integer.

The last thing we need is a second conjecture, which has been proved:

Conjecture $2$: For any positive integer $n$, there exist at least $1$ prime $p>n$ with the digits of $n$ as first digits.

See: Proof that there are infinitely many prime numbers starting with a given digit string.

Proof:

Let $n$ be a type $α$ integer. Because of fact $2$, any prime $p$ with the digits of $n$ as first digits is either a type $β$ integer $>n$, or an integer with a type $β$ initial segment $>n$. Therefore, since conjecture $2$ is true, for any type $α$ integer, there exists a greater type $β$ integer. Because of fact $1$, we can conclude that there are infinitely many type $β$ integers.

François Huppé
  • 1,252