The conjecture was formulated by Rajat Bhattacharjee and Prashant Pandey in their 2001 thesis. It has been computationally verified for r and n , and for r = 5, n . However, a heuristic argument by
Carl Pomerance and
Hendrik W. Lenstra suggests there are infinitely many counterexamples. In particular, the heuristic shows that such counterexamples have asymptotic density greater than \tfrac{1}{n^{\varepsilon}} for any \varepsilon > 0. Assuming Agrawal's conjecture is false by the above argument, Roman B. Popovych conjectures a modified version may still be true: Let n and r be two coprime positive integers. If :(X - 1)^n \equiv X^n - 1 \pmod{n,\, X^r - 1} and :(X + 2)^n \equiv X^n + 2 \pmod{n,\, X^r - 1} then either n is prime or n^2 \equiv 1 \pmod{r}. == Distributed computing ==