# When Is the Digital Root Three?

Number Theory Level 4

For how many positive integers $$n \leq 1000$$, is the digit root of $$n^2 + n +1$$ equal to 3?

Details and assumptions

The digit root of a number is a single digit value obtained by iterative digit sums. For example, the digit root of $$31^2+31+1 = 993$$ would be 3 since $$9 + 9 + 3 = 21, 2+1 = 3$$.

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