Enter a positive integer to check if it is a Carmichael number using Korselt criterion
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Result
Verdict
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Detailed Derivation
Carmichael Number Properties
Korselt Criterion: n is Carmichael iff
1. n is composite and square-free
2. For each prime p|n: (p-1) | (n-1)
Smallest: 561 = 3 x 11 x 17
Carmichael numbers are composite numbers that behave like primes under the Fermat primality test. They are rare but infinitely many exist, making them important in computational number theory and cryptography.
⚠Results are best for numbers under 10^9. Larger numbers may take time to factor. The tool checks Korselt criterion via prime factorization.
What Is a Carmichael Number?
A Carmichael number is a composite integer that passes the Fermat test for all bases coprime to it. These deceptively prime-like composites are important in the study of primality testing and cryptographic security.
Square-Free
Carmichael numbers have no squared prime factors. Each prime appears with exponent 1 in the factorization.
Fermat Liars
They are Fermat liars for every base. For all a with gcd(a,n)=1, a^(n-1) = 1 (mod n). Normal composites only pass for some bases.
Infinitely Many
Proved infinitely many in 1994. The growth rate is more than n^(1/3). Over 1.4 million exist below 10^18.
Cryptography
RSA key generation uses Miller-Rabin test instead of Fermat because Carmichael numbers would pass the simpler test.
Teaching Example: Test 561. Factor: 561 = 3 x 11 x 17. Square-free? Yes. Check: (3-1)=2 divides 560? Yes. (11-1)=10 divides 560? Yes. (17-1)=16 divides 560? Yes. 561 is Carmichael!
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