Given integers a, b, c, d, consider the arithmetic progression N = ak + b.
If the index k can be expressed in the form: k = axy + cx + dy for some integers x and y,
AND if the parameters a, b, c, d and the variables x, y satisfy the condition: cd – b = a(c – d)(x – y)
THEN ak + b can be factored as: ak + b = (ax + c)(ay + d)
Consequently, N = |ak + b| = |(ax + c)(ay + d)| will be composite if |ax + c| > 1 and |ay + d| > 1.
Key Implications:
Sieve Mechanism: This provides a mechanism to identify indices k that guarantee ak+b is composite due to a specific type of factorization.
Conditional Factoring: The factorization is conditional upon the relationship cd – b = a(c-d)(x-y) holding true for the chosen a,b,c,d and the specific x,y that generate k.
Simplified Case (Unconditional for k form): If c = d, the condition simplifies to b = c².
In this scenario, for any k of the form axy + cx + cy, the expression ak + c² will always factor as (ax + c)(ay + c).
This formulation directly links the specific form of k and the side condition to the resulting factorization.