If $A = \{5p+7q|p,q \in \mathbb{N}\}$, determine $|\mathbb{N} \setminus A|$.
2026-03-31 03:39:09.1774928349
How many natural numbers can't be written as 5p+7q, where p and q are natural numbers?
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For two prime numbers p,q the set which can not be represented as pa+qb with $a,b \in N$ is finite. There is a maximal non-representable number which is pq-p-q. This is known as the Frobenius number. There is a formula for the number of non-representable numbers $\frac{(p-1)(q-1)}{2}$.
Refs:
Ramírez-Alfonsín, J. L. The Diophantine Frobenius Problem. Oxford, England: Oxford University Press, 2005.