|
%I #20 Feb 24 2022 02:06:36
%S 5,145,205,725,1025,1105,1145,1205,1305,1313,1365,1405,1469,1745,1785,
%T 1845,1885,1989,2145,2249,2405,2465,2545,2665,2745,2805,3005,3045,
%U 3145,3161,3205,3393,3445,3485,3545,3601,3625,3705,3885,3893,3965
%N Numbers m == 1 (mod 4) such that the Gauss factorial ((m-1)/4, m)! == 1 (mod m).
%C The Gauss factorial m_k! is defined to be Product_{1<=j<=m, gcd(j,k)=1} j.
%H J. B. Cosgrave and K. Dilcher, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.09.812">An introduction to Gauss factorials</a>, Amer. Math. Monthly, 118 (2011), 810-828.
%H J. B. Cosgrave and K. Dilcher, <a href="http://dx.doi.org/10.1007/s10474-013-0357-1">The Gauss-Wilson theorem for quarter-intervals</a>, Acta Mathematica Hungarica, Sept. 2013.
%e m=145 is a term, because 36_145! = 32577412307818387955599294857216 == 1 (mod 145).
%p Gf:=proc(N,n) local j,k; k:=1;
%p for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
%p t1:=[];
%p for i from 1 to 1000 do
%p n:=4*i+1; if (Gf(i,n) mod n ) = 1 then t1:=[op(t1),n]; fi;
%p od:
%p t1;
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Dec 08 2013
|
