Consider two 5-digit perfect squares, the first two digits of each of which form a 2-digit prime number, and the last three digits form a 3-digit prime number.

For sake of discussion, let the digits be called ABCDE and VWXYZ. The two squares I'm thinking of can form, from those digits, another 5-digit square: ABXYZ. It is of the same type as the other two as AB is prime as is XYZ.

The use of different letters does not imply that all the letters represent different digits; any two may be the same or different, but the combined square does share its first two digits with those of one of the two original squares and its last three with the last three of the other.

What are the three squares?

I used mathematica to find the following solutions for the 3 squares

11449 , 11449 , 11449
 11449 , 11881 , 11881
 11449 , 19881 , 11881
 11449 , 23409 , 11409
 11449 , 29241 , 11241
 11449 , 29929 , 11929
 11449 , 83521 , 11521
 11449 , 89401 , 11401
 11881 , 11449 , 11449
 11881 , 11881 , 11881
 11881 , 19881 , 11881
 11881 , 23409 , 11409
 11881 , 29241 , 11241
 11881 , 29929 , 11929
 11881 , 83521 , 11521
 11881 , 89401 , 11401
 19881 , 11449 , 19449
 19881 , 11881 , 19881
 19881 , 19881 , 19881
 19881 , 23409 , 19409
 19881 , 29241 , 19241
 19881 , 29929 , 19929
 19881 , 83521 , 19521
 19881 , 89401 , 19401
 23409 , 11449 , 23449
 23409 , 11881 , 23881
 23409 , 19881 , 23881
 23409 , 23409 , 23409
 23409 , 29241 , 23241
 23409 , 29929 , 23929
 23409 , 83521 , 23521
 23409 , 89401 , 23401
 29241 , 11449 , 29449
 29241 , 11881 , 29881
 29241 , 19881 , 29881
 29241 , 23409 , 29409
 29241 , 29241 , 29241
 29241 , 29929 , 29929
 29241 , 83521 , 29521
 29241 , 89401 , 29401
 29929 , 11449 , 29449
 29929 , 11881 , 29881
 29929 , 19881 , 29881
 29929 , 23409 , 29409
 29929 , 29241 , 29241
 29929 , 29929 , 29929
 29929 , 83521 , 29521
 29929 , 89401 , 29401
 83521 , 11449 , 83449
 83521 , 11881 , 83881
 83521 , 19881 , 83881
 83521 , 23409 , 83409
 83521 , 29241 , 83241
 83521 , 29929 , 83929
 83521 , 83521 , 83521
 83521 , 89401 , 83401
 89401 , 11449 , 89449
 89401 , 11881 , 89881
 89401 , 19881 , 89881
 89401 , 23409 , 89409
 89401 , 29241 , 89241
 89401 , 29929 , 89929
 89401 , 83521 , 89521
 89401 , 89401 , 89401

And the code I used to find these follows

For[a=1,a„T9,a++,

For[b=0,b„T9,b++,

n1=10*a+b;

If[PrimeQ[n1],

For[c=1,c„T9,c++,

For[d=0,d„T9,d++,

For[e=0,e„T9,e++,

n2=100*c+10*d+e;

n3=1000*n1+n2;

If[PrimeQ[n2] && IntegerQ[Sqrt[n3]],

For[v=1,v„T9,v++,

For[w=0,w„T9,w++,

n4=10*v+w;

If[PrimeQ[n4],

For[x=1,x„T9,x++,

For[y=0,y„T9,y++,

For[z=0,z„T9,z++,

n5=100*x+10*y+z;

n6=1000*n4+n5;

n7=1000*n1+n5;

If[PrimeQ[n5] && IntegerQ[Sqrt[n6]] && IntegerQ[n7],

Print[n3,",",n6,",",n7];

];

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Posted by Daniel on 2009-04-09 11:42:29