Agrawal's conjecture

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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 n and r be two coprime positive integers. If

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

then either n is prime or n^2 \equiv 1 \pmod r

Ramifications[edit]

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

Truth or falsehood[edit]

Agrawal's conjecture has been computationally verified for r < 100 and n < 10^{10}, however a heuristic argument by Carl Pomerance and Hendrik W. Lenstra suggests there is an infinite number of counterexamples.[2] In particular, the heuristic shows that such counterexamples have asymptotic density greater than \tfrac{1}{n^{\epsilon}} for any \epsilon > 0.

Assuming Agrawal's conjecture is false by the above argument, a modified version (the Agrawal–Popovych conjecture[3]) 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}.

Notes[edit]

  1. ^ Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (2004). "PRIMES is in P". Annals of Mathematics 160 (2): 781–793. doi:10.4007/annals.2004.160.781. JSTOR 3597229. 
  2. ^ Lenstra, H. W.; Pomerance, Carl. "Remarks on Agrawal’s conjecture.". American Institute of Mathematics. Retrieved 16 October 2013. 
  3. ^ Popovych, Roman, A note on Agrawal conjecture