The Wikipedia article lists the first happy numbers as 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, ..., so the answer is clearly no.
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A positive number added to another positive number always results in a positive sum.
A negative plus a negative still gives a negative answer.
Because the sum of two odd numbers is always even and the sum of that even number plus another odd number is always odd.
No.
Yes.