Agrawal's conjecture

In number theory, Agrawal's conjecture, due to Manindra Agrawal in 2002,^[1] forms the basis for the cyclotomic AKS test. Agrawal's conjecture states formally:

Let ${\displaystyle n}$ and ${\displaystyle r}$ be two coprime positive integers. If

${\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,\,X^{r}-1}}\,}$

then either ${\displaystyle n}$ is prime or ${\displaystyle n^{2}\equiv 1{\pmod {r}}}$

Ramifications

If Agrawal's conjecture were true, it would decrease the runtime complexity of the AKS primality test from ${\displaystyle {\tilde {O}}{\mathord {\left(\log ^{6}n\right)}}}$ to ${\displaystyle {\tilde {O}}{\mathord {\left(\log ^{3}n\right)}}}$.

Truth or falsehood

The conjecture was formulated by Rajat Bhattacharjee and Prashant Pandey in their 2001 thesis.^[2] It has been computationally verified for ${\displaystyle r<100}$ and ${\displaystyle n<10^{10}}$,^[3] and for ${\displaystyle r=5,n<10^{11}}$.^[4]

However, a heuristic argument by Carl Pomerance and Hendrik W. Lenstra suggests there are infinitely many counterexamples.^[5] In particular, the heuristic shows that such counterexamples have asymptotic density greater than ${\displaystyle {\tfrac {1}{n^{\varepsilon }}}}$ for any ${\displaystyle \varepsilon >0}$.

Assuming Agrawal's conjecture is false by the above argument, Roman B. Popovych conjectures a modified version may still be true:

Let ${\displaystyle n}$ and ${\displaystyle r}$ be two coprime positive integers. If

${\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,\,X^{r}-1}}}$

and

${\displaystyle (X+2)^{n}\equiv X^{n}+2{\pmod {n,\,X^{r}-1}}}$

then either ${\displaystyle n}$ is prime or ${\displaystyle n^{2}\equiv 1{\pmod {r}}}$.^[6]

Distributed computing

Both Agrawal's conjecture and Popovych's conjecture were tested by distributed computing project Primaboinca which ran from 2010 to 2020, based on BOINC. The project found no counterexample, searching in ${\displaystyle 10^{10}<n<10^{17}}$.