Numbers Which Are Both Squares and Cubes Must be of Form 7k, 7k+1
\[x\]
is both a square and a cube. Then we can write \[x=m^2=n^3\]
. Only numbers of certain forms can be both a square and a cube. The table shows such numbers modulo 7.\[m, \; n\] | \[m^2\] | \[n^3\] |
0 | \[0 \equiv 0 \; (mod \; 7)\] | \[0 \equiv 0 \; (mod \; 7)\] |
1 | \[1 \equiv 1 \; (mod \; 7)\] | \[1 \equiv 1 \; (mod \; 7)\] |
2 | \[4 \equiv 4 \; (mod \; 7)\] | \[8 \equiv 1 \; (mod \; 7)\] |
3 | \[9 \equiv 2 \; (mod \; 7)\] | \[27 \equiv 6 \; (mod \; 7)\] |
4 | \[16 \equiv 2 \; (mod \; 7)\] | \[64 \equiv 1 \; (mod \; 7)\] |
5 | \[25 \equiv 4 \; (mod \; 7)\] | \[125 \equiv 6 \; (mod \; 7)\] |
6 | \[36 \equiv 1 \; (mod \; 7)\] | \[216 \equiv 6 \; (mod \; 7)\] |
7 | \[49 \equiv 0 \; (mod \; 7)\] | \[343 \equiv 0 \; (mod \; 7)\] |
8 | \[64 \equiv 1 \; (mod \; 7)\] | \[512 \equiv 1 \; (mod \; 7)\] |
9 | \[81 \equiv 4 \; (mod \; 7)\] | \[729 \equiv 1 \; (mod \; 7)\] |
\[7k\]
or \[7k_1\]
.