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%I #22 Jul 02 2022 01:51:05
%S 588107520,67270694400,546939993600,2128050512640,37400697734400,
%T 5566067918611200
%N Numbers expressible as (a^2 - 1)*(b^2 - 1) in 5 distinct ways.
%C The next term (if it exists) is greater than 2^70.
%H K. S. Brown, <a href="http://www.mathpages.com/home/kmath275.htm">Numbers Expressible as (a^2-1)(b^2-1)</a>
%H K. S. Brown, <a href="http://www.mathpages.com/home/mwlist.htm">MathPages Most Wanted List</a>, Problem #14.
%o (PARI) { f(a,b) = (a+1)*(a-1)*(b+1)*(b-1) } ans=vector(6,x,[0,0]); clear=ans; { g(a) = b=divisors(a*a-1); l=length(b); b=b+vector(l,x,a); for(x=1,l/2,c=4*a*b[x]*(a+1)*(a-1)*(b[x]+1)*(b[x]-1)*(a*b[x]-1)/((b[x]-a)*(b[x]-a));
%o d=floor(sqrt(sqrt(c))); count=1; for( y=2,d, if (c%(y*y-1)==0,e=ceil(sqrt(c/(y*y-1))); if (f(y,e)==c,ans[count]=[y,e]; count=count+1,),); ); if ( count>5,print("g:",a," ",c," ",ans); ans=clear,); ); } { find()= for(n=560,10001,print(n); g(n)); }
%o Store program as text file, load gp, \r textfilename and then run function find() to search for a 7th entry.
%o (PARI) { f(a,b) = (a+1)*(a-1)*(b+1)*(b-1) } ans=vector(6,x,[0,0])
%Y Cf. A063066, A063067, A063068.
%Y Cf. A134856, A134857, A134858 (identical?).
%K nonn
%O 1,1
%A K. S. Brown (ksbrown(AT)seanet.com), Fred W. Helenius (fredh(AT)ix.netcom.com), _Dean Hickerson_, _Randall L Rathbun_