If Agrawal's conjecture were true, it would decrease the runtime complexity of the AKS primality test from to .
Truth or falsehoodedit
The conjecture was formulated by Rajat Bhattacharjee and Prashant Pandey in their 2001 thesis.[2] It has been computationally verified for and ,[3] and for .[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 for any .
Assuming Agrawal's conjecture is false by the above argument, Roman B. Popovych conjectures a modified version may still be true:
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 .
Notesedit
^Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (2004). "PRIMES is in P" (PDF). Annals of Mathematics. 160 (2): 781–793. doi:10.4007/annals.2004.160.781. JSTOR 3597229.
^Saxena, Nitin (Dec 2014). "Primality & Prime Number Generation" (PDF). UPMC Paris. Archived from the original (PDF) on 25 April 2018. Retrieved 24 April 2018.
^Lenstra, H. W.; Pomerance, Carl (2003). "Remarks on Agrawal's conjecture" (PDF). American Institute of Mathematics. Retrieved 16 October 2013.
^
Popovych, Roman (30 December 2008), A note on Agrawal conjecture(PDF), retrieved 21 April 2018