Extending Goldbach's Conjecture

Let $m$ be a positive integer. Writing an even $m$ of at least $4$ as a sum of two primes, and writing an odd $m$ greater than $7$ as a sum of three primes, are questions that no algorithm settles today, and nobody has proved that such a sum always exists.

So we change the target. Numbers that add up to $m^3$ are more likely to exist than numbers that add up to $m$. To keep the search from turning trivial and the checking from turning hard (a sum of primes can have many answers), we look for $m$ consecutive odd integers instead of primes. When such a run exists, it is the only one.

For example, take $m = 2$. Then $2^3 = 8$, and $8$ is the sum of the two consecutive odd integers $3 + 5$.

Each line holds one integer $m$ with $1 \le m \le 10^9$. The last line holds $m = 0$ and marks the end of the input. The input has at most $100$ values of $m$.

For each $m$ before the terminating $0$, print one line holding two fields $A$ and $B$ separated by a single space.

$A$ is Y if $m$ consecutive odd integers adding up to $m^3$ exist, and N if they do not.

$B$ is the first of those $m$ odd integers when $A$ is Y, and $0$ otherwise.