Are there an infinite number of lucky primes?

Definitions and mathematical statement

Begin with the increasing sequence $L_1=(1,3,5,7,9,\ldots)$ obtained by deleting every second entry from the positive integers. For each integer $j\ge2$, let $\ell_j$ be the $j$th entry of the increasing sequence $L_{j-1}$. Form $L_j$ by deleting entries whose positions in $L_{j-1}$ are multiples of $\ell_j$, with positions counted from one and all deletions in that stage determined simultaneously. Also put $\ell_1=1$.

The lucky numbers are the positive integers surviving every stage, equivalently the increasing sequence $\ell_1,\ell_2,\ldots$. Its beginning is \[ 1,3,7,9,13,15,21,25,31,\ldots. \] After the initial deletion of even positions, the successive sieve step sizes are therefore $3,7,9,\ldots$, chosen from the surviving sequence itself. They are not the consecutive integers $3,4,5,\ldots$ and are not necessarily prime.

A lucky prime is a lucky number that is prime in the ordinary multiplicative sense: it is greater than one and has no positive divisors other than one and itself. The conjecture asks whether there are infinitely many lucky primes. Equivalently, for every positive integer $B$ there should exist a surviving number $\ell>B$ with this primality property.

Infinitely many lucky numbers alone do not establish the conjecture, since surviving the positional sieve is not a divisibility test. The source's prose mixes these sieve conventions; the recursive definition above fixes the standard lucky-number sequence intended by its title and discussion.

Short English statement

Are infinitely many numbers surviving the standard lucky-number positional sieve ordinary prime numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3365 (OPG-37192), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Are there an infinite number of lucky primes?. Original problem and discussion. http://www.openproblemgarden.org/op/are_there_an_infinite_number_of_lucky_primes.
• S3 C. Sanna, Explicit inequalities for the nth lucky number (2026). https://arxiv.org/abs/2604.07142.