[number theory] sieving with a_i mod p_i

[jaydfox]

Thank you for your reply and intrest Jay !

However Im not sure what you computed.
(...)
But then you write

a_i
..
28..33

(...)

Does that mean that for any of 28,29,... 33 as a fixed value of a_i we get the same result ?

Many interpretations are possible I think.

Im betting you are considering a fixed a_i and

0..27 1204 means that for any fixed a_i between 0 and 27 we get the same value : 1204.

Is my guess correct ?

Exactly what I meant! For a fixed a_i of 0 (or 1, or 2, or 27, etc.), I found 1204 "primes" in the range of 101 to 10000 inclusive.

Note that in my OP - which appears confusing apparantly - the a_i are not neccessarily fixed.

So we could consider
1 mod 2 ,1 mod 3, 2 mod 5, 6 mod 7, 5 mod 11 etc

I assume fixed a_i are easier to study perhaps.

Well, my modular arithmetic is rusty, but it seems to me that {1 mod 2, 1 mod 3, 2 mod 5, 6 mod 7, 5 mod 11}, as a set, defines a unique number mod 2*3*5*7*11. Assuming my educated guess is right, you basically are fixing a_i, though not limited to the same range as initially defined. So I think it still makes sense to speak in terms of a fixed a_i, if perhaps we relax the restriction on the size of a_i (up to primorial(p_i)-1 or something like that).