If S is NOT divisible by 9, then we observe that:
S ≡ 1,4,7(mod 9)
=> sod (S) ≡ 1,4,7 (mod 9)
Hence:
S+sod(S) ≡ 2, 5, 8 (mod 9).
However, 2,5,8 are not quadratic residues in the mod 9 system.
Therefore, S+ sod(S) can never be a perfect square,
whenever S is NOT divisible by 9. Let S'= S/(9^x). Of course, for some value of x>=1, it follows that
S'=S/(9^x) is NOT divisible by 9.
Then, by previous arguments, it follows that:
Therefore, it follows that S+sod(S) is NOT a perfect
square whenever S is divisible by 9, except when S=0
Consequently, except for the trivial S=0, there does NOT exist squares S = n^2 such that when you add S and its digits the result is also a square. |
