Alexa's Conjecture on Primality

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For each integer $p\ge8$, put $k(p)=\lfloor p^{1/3}/2\rfloor$. For $1\le i\le k(p)$, let $r_i(p)$ be the least nonnegative residue of $(2i+1)^{p-1}$ modulo $p$, so that $0\le r_i(p)<p$ and $r_i(p)\equiv(2i+1)^{p-1}\pmod p$.

Statement recorded in the supplied source.

Is the following equivalence valid for every integer $p\ge8$? \[ p\text{ is prime}\quad\Longleftrightarrow\quad \sum_{i=1}^{k(p)}r_i(p)=k(p). \] The displayed equality is an equality of integers, and each residue is chosen separately before summing. [S2]

Short English statement

Determine whether this finite sum of specified Fermat residues characterizes primes at least eight.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3361 (OPG-37413). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Alexa's Conjecture on Primality, OPG-37413. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/alexas_conjecture_on_primality.