CoCalc Public Fileselementary Carmichael example.sagews
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Compute Environment: Ubuntu 18.04 (Deprecated)
# In this worksheet, we will test to see if
# 172081 is a Carmichael number by applying Korselt's criterion
# and only a few basic commands (factor and Mod)

# Korselt: We need to check that (1) n is composite and squarefree
# and (2) that for every prime p dividing n, that (p-1) divides (n-1)

# To check (1), we look at the factorization of n:
n=172081
factor(n)

7 * 13 * 31 * 61
# Hence (1) is satisfied: we can see that every factor is only to the 1st power, so n is squarefree

# To check condition (2), we will show (p-1) divides (n-1) by checking that (n-1) is 0 modulo (p-1)
m=n-1
Mod(m, 6)

0
Mod(m,12)

0
Mod(m,30)

0
Mod(m,60)

0
# Hence (2) is satisfied: p-1 does indeed divide n-1 for all primes p dividing n

# We conclude that 172080 is a Carmichael number.


# In this worksheet, we will test to see if
# 172081 is a Carmichael number by applying Korselt's criterion
# and only a few basic commands (factor and Mod)

# Korselt: We need to check that (1) n is composite and squarefree
# and (2) that for every prime p dividing n, that (p-1) divides (n-1)

# To check (1), we look at the factorization of n:
n=172081
factor(n)

7 * 13 * 31 * 61
# Hence (1) is satisfied: we can see that every factor is only to the 1st power, so n is squarefree

# To check condition (2), we will show (p-1) divides (n-1) by checking that (n-1) is 0 modulo (p-1)
m=n-1
Mod(m, 6)

0
Mod(m,12)

0
Mod(m,30)

0
Mod(m,60)

0
# Hence (2) is satisfied: p-1 does indeed divide n-1 for all primes p dividing n

# We conclude that 172080 is a Carmichael number.