%I A053200
%S A053200 0,0,0,1,0,1,1,0,0,1,1,0,2,0,1,1,0,0,0,0,1,1,0,3,2,3,0,1,1,0,0,0,0,0,0,
%T A053200 1,1,0,4,0,6,0,4,0,1,1,0,0,3,0,0,3,0,0,1,1,0,5,0,0,2,0,0,5,0,1,1,0,0,0,
%U A053200 0,0,0,0,0,0,0,1,1,0,6,4,3,0,0,0,3,4,6,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A053200 Binomial coefficients C(n,k) reduced modulo n, read by rows ; T(0,0)=0 
               by convention .
%C A053200 Pascal's triangle read by rows, where row n is read mod n.
%C A053200 A number n is a prime if and only if (1+x)^n == 1+x^n (mod n), i.e. if 
               and only if the n-th row is 1,0,0,...,0,1. This result underlies 
               the proof of Agrawal, Kayal and Saxena that there is polynomial-time 
               algorithm for primality testing. - N. J. A. Sloane (njas(AT)research.att.com), 
               Feb 20, 2004
%D A053200 M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Ann. of Math. (2) 
               160 (2004), no. 2, 781-793.
%H A053200 T. D. Noe, <a href="b053200.txt">Rows n=0..100 of triangle, flattened</
               a>
%e A053200 Row 4 = 1 mod 4, 4 mod 4, 6 mod 4, 4 mod 4, 1 mod 4 = 1, 0, 2, 0, 1
%e A053200 Triangle begins :
%e A053200 0 ;
%e A053200 0,0 ;
%e A053200 1,0,1 ;
%e A053200 1,0,0,1 ;
%e A053200 1,0,2,0,1 ;
%e A053200 1,0,0,0,0,1 ;
%e A053200 1,0,3,2,3,0,1 ;
%e A053200 1,0,0,0,0,0,0,1 ;
%e A053200 1,0,4,0,6,0,4,0,1 ;
%e A053200 1,0,0,3,0,0,3,0,0,1 ;
%e A053200 1,0,5,0,0,2,0,0,5,0,1 ;
%e A053200 1,0,0,0,0,0,0,0,0,0,0,1 ;
%e A053200 1,0,6,4,3,0,0,0,3,4,6,0,1 ;
%e A053200 1,0,0,0,0,0,0,0,0,0,0,0,0,1 ;
%p A053200 f := n -> seriestolist( series( expand( (1+x)^n ) mod n, x, n+1)); (N. 
               J. A. Sloane)
%Y A053200 Row sums give A053204. Cf. A053201, A053202, A053203, A007318 (Pascal's 
               triangle)
%Y A053200 Cf. also A092241.
%Y A053200 Cf. A007318.
%Y A053200 Sequence in context: A037273 A158924 A025426 this_sequence A050870 A103306 
               A163510
%Y A053200 Adjacent sequences: A053197 A053198 A053199 this_sequence A053201 A053202 
               A053203
%K A053200 nonn,tabl,nice
%O A053200 0,13
%A A053200 Asher Auel (asher.auel(AT)reed.edu) Dec 12, 1999
%E A053200 Corrected by T. D. Noe, Feb 08 2008
%E A053200 Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 29 2008 at 
               the suggestion of R. J. Mathar