A057535 Numbers expressible as (a^2 - 1)*(b^2 - 1) in 5 distinct ways.

588107520, 67270694400, 546939993600, 2128050512640, 37400697734400, 5566067918611200

Offset: 1

Author

K. S. Brown (ksbrown(AT)seanet.com), Fred W. Helenius (fredh(AT)ix.netcom.com), Dean Hickerson, Randall L Rathbun

Keywords

• nonn

Comments

The next term (if it exists) is greater than 2^70.

Links

• K. S. Brown, Numbers Expressible as (a^2-1)(b^2-1)
• K. S. Brown, MathPages Most Wanted List, Problem #14.

Crossrefs

Cf. A063066, A063067, A063068.

Cf. A134856, A134857, A134858 (identical?).

Programs

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));
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)); }
Store program as text file, load gp, \r textfilename and then run function find() to search for a 7th entry.

PARI

{ f(a,b) = (a+1)*(a-1)*(b+1)*(b-1) } ans=vector(6,x,[0,0])