Is (10^y - 1) x (10^y) + 1 prime for any value of y besides 2, 4, 6, and 8?

[Don Reble]

> ... an obvious reason why (10^y - 1) x (10^y) + 1
> could only produce primes when y is 2, 4, 6, or 8?

Let f(y) = (10^y - 1) * 10^y + 1. One can prove:

    If y is odd but not a multiple of 9,
    then f(y) is divisible by 7 or 19;

    If y is even but not a multiple of 6,
    let y=2^k*x (odd x, maximum k);
    then f(2^k) divides f(y).

So if f(y) is prime, then
    y is a power of two, or y == 0,6,9,12 mod 18.

-- 
Don Reble  djr at nk.ca



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