A090659 Odd composites with increasing proportion of nontrivial non-witnesses of compositeness by the Miller-Rabin primality test.

25, 91, 703, 1891, 12403, 38503, 79003, 88831, 146611, 188191, 218791, 269011, 286903, 385003, 497503, 597871, 736291, 765703, 954271, 1056331, 1314631, 1869211, 2741311, 3270403, 3913003, 4255903, 4686391, 5292631, 6186403, 6969511, 8086231, 9080191

Offset: 1

Author

Ken Takusagawa, Dec 14 2003

Keywords

• nonn

Comments

Rabin has shown that the proportion has an upper bound of 0.25. If the trivial non-witnesses are counted, this upper bound is reached at 9. If the conjecture is true that the later terms are always the product of two primes p and (2*p-1), then the sequence continues 188191 218791 269011 286903 385003 497503 597871 736291 765703 954271 1056331 1314631 1869211 2741311 3270403 3913003 4255903 4686391 5292631.

Dickson's conjecture implies that this sequence is infinite. Can this be proved unconditionally? - Charles R Greathouse IV, Mar 10 2011

Higgins' conjecture 2 is implied by his conjecture 1, which is true by the general form of the number of non-witnesses of an odd number. - Charles R Greathouse IV, Mar 10 2011

Examples

			25 has 2 nontrivial non-witnesses (NTNW), namely (7,18), for a proportion of 2/22=0.0909. The denominator is 22 because the non-witnesses are selected from 2..23 (as 1 and 24 are trivial non-witnesses).
49 has 4 NTNW, namely (18,19,30,31) for a proportion of 4/46=0.0870. This is a smaller proportion than 0.0909 for 25.
91=7*13 has 16 NTNW in the range [2..89], namely [9, 10, 12, 16, 17, 22, 29, 38, 53, 62, 69, 74, 75, 79, 81, 82], for a proportion of 16/88=0.182. It also has two trivial non-witnesses 1 and 90, which are not counted. The next integer with a higher proportion is 703, with 160 nontrivial non-witnesses and proportion 0.229.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..5411
• Brian C. Higgins, The Rabin-Miller Primality Test: Some Results on the Number of Non-witnesses to Compositeness, ALLEMO Spring 1996 Meeting at IUP, Proceedings Volume 1.
• S. Narayanan, Improving the Speed and Accuracy of the Miller-Rabin Primality Test, MIT PRIMES-USA, 2015.
• Michael O. Rabin, Probabilistic algorithm for testing primality, Journal of Number Theory 12:1 (1980), pp. 128-138.

Crossrefs

Subsequence of A141768.

Extensions

Extended and edited by Charles R Greathouse IV, Mar 09 2011