Looking for a smaller solution...

[wraithx@morpheus.net]

Hello,

While I was working on a problem, someone offered a Pari/GP solution.  While it 
works great, the solution is larger than I'd like.  I was wondering if there was 
an alternate way to compute the solution that results in a smaller answer?

Here is a simplified version of the problem I'm working on:
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Does anyone know of a way to find, or construct, a number B = k*x + y with x and 
y given, that is only divisible by primes from a finite set of primes P?

For example, if x=97929 and y=83261 and P = [1171, 1429, 1811, 2339], how would 
we find or construct B? Or, perhaps show that one doesn't exist?

I know we could do a brute force search, looping over k values and checking the 
factors of B, but I'm hoping someone might know of a way to construct, or 
constructively find, B. Is this possible?
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In the full problem I'm working on, my numbers x and y have a few hundred 
digits, and my set P has over 1000 primes.  Also, x has only small prime factors 
p^e, with p a prime < 1000 and where e is usually 1, but can be higher, like 
2^13, or 3^8, 7^5, etc. Also, all p are less than the smallest prime in P, ie no 
prime is in both sets.

The code that was offered is:
b(x,y,P)={G=znstar(x,1);matsolvemod(matconcat(vector(#P,i,znlog(P[i],G))),G.cyc~,znlog(y,G))}
which "outputs a vector giving the exponents that each of the primes in P need 
to be raised to in the product."  ie B = product(P[i]^e[i]) where e[i] is the 
output of the above function.

For the full problem, this does give a solution, but each exponent e[i] in the 
solution has over 90 digits.  Also, Pari/GP only finds exponents for 142 of the 
primes from P (the rest of the exponents for the other primes in the set are 0). 
 I understand this is due to the fact that my x value has 141 factors (p_i^e_i).

Does anyone know of a way to reduce the size of the "exponents" in the solution? 
 Perhaps by using more primes from the set P?  Is there a way to get Pari/GP to 
use more than 142 primes from P to generate its solution to this problem?  I 
don't need to find the smallest B value to my problem, but I would like to find 
a "small" B value that is computable.  Perhaps a B value with only a few 
hundred, or a few thousand, digits.  Thank you in advance for any help!

-David C.