The abstract theory centers on Carmichael numbers within number theory, defined as composite integers \(n\) where every integer relatively prime to \(n\) satisfies the congruence condition \(a^{n-1} \equiv 1 \pmod{n}\). The core mechanism involves constructing these numbers via specific sets of prime factors and applying analytic techniques derived from studying small gaps between standard primes, specifically leveraging sieve theory concepts associated with Bombieri-Pomerance. This theoretical framework establishes bounds for the distribution of Carmichael numbers in intervals \(n\) to \(2^n\), addressing their role as deceptive pseudoprimes that compromise cryptographic primality testing by mimicking prime behavior under modular exponentiation without being prime themselves.
The abstract theory centers on Carmichael numbers within number theory, defined as composite integers \(n\) where every integer relatively prime to \(n\) satisfies the congruence condition \(a^{n-1} …