Determine whether 561 is a Carmichael number, and state the general criterion that decides it.
It has been determined that 561 is a Carmichael number. The general criterion for this, Korselt's Criterion, was stated and applied. Specifically, it was shown that:
1. 561 is a composite number with prime factorization 3 x 11 x 17.
2. 561 is square-free.
3. For each prime factor p (3, 11, 17), (p-1) divides (561-1).
def prime_factors(n):
factors = []
d = 2
while d * d <= n:
while (n % d) == 0:
factors.append(d)
n //= d
d += 1
if n > 1:
factors.append(n)
return factors
print(prime_factors(561))
[3, 11, 17]
n = 561
factors = []
d = 2
temp_n = n
while d * d <= temp_n:
if temp_n % d == 0:
factors.append(d)
temp_n //= d
else:
d += 1
if temp_n > 1:
factors.append(temp_n)
print(f'Prime factorization of {n}: {factors}')
Prime factorization of 561: [3, 11, 17]
n = 561
factors = [3, 11, 17]
is_square_free = len(factors) == len(set(factors))
print(f'Is {n} square-free? {is_square_free}')
Is 561 square-free? True
n = 561
factors = [3, 11, 17]
n_minus_1 = n - 1
for p in factors:
p_minus_1 = p - 1
result = n_minus_1 // p_minus_1
remainder = n_minus_1 % p_minus_1
if remainder == 0:
print(f'For p = {p}, (p-1) = {p_minus_1}. {n_minus_1} / {p_minus_1} = {result}.')
For p = 3, (p-1) = 2. 560 / 2 = 280. For p = 11, (p-1) = 10. 560 / 10 = 56. For p = 17, (p-1) = 16. 560 / 16 = 35.
Acceptance criteria set:
1. The formal definition of a Carmichael number is stated: a composite number n such that for all integers b with gcd(b, n) = 1, the congruence b^(n-1) ≡ 1 (mod n) holds.
2. Korselt's Criterion is stated as the general decision procedure, specifying that a composite number n is a Carmichael number if and only if it is square-free and for every prime factor p of n, (p-1) divides (n-1).
3. The complete prime factorization of 561 is provided as 3 × 11 × 17, demonstrating that it is a composite number.
4. It is verified that 561 is square-free, justified by the fact that its prime factorization contains no repeated prime factors.
5. The divisibility condition of Korselt's Criterion is demonstrated for each prime factor of 561, showing the explicit integer results of the divisions: 560/(3-1), 560/(11-1), and 560/(17-1).
6. A final conclusion is stated, declaring that 561 is a Carmichael number because it has been shown to satisfy all conditions of Korselt's Criterion.
The search results mention Carmichael numbers but don't provide a clear, quotable definition. Let me search for a more authoritative source with the formal definition.
Let me read the Wikipedia article for a formal, citable definition:
This Wikipedia article provides the formal definition. Let me record and cite this properly.
Criterion 1 is now addressed with a proper citation from Wikipedia. The formal definition of a Carmichael number has been stated:
A Carmichael number is a composite number n such that for all integers b with gcd(b, n) = 1, the congruence b^(n-1) ≡ 1 (mod n) holds.
This is cited from Wikipedia's article on Carmichael numbers, which states both the general form (b^n ≡ b (mod n) for all integers b) and the equivalent form using relative primality (b^(n-1) ≡ 1 (mod n) for all b coprime to n).
Addressing criterion 1 with checkable evidence.
-
Recorded fact: A Carmichael number is a composite number n such that for all integers b with gcd(b, n) = 1, the congruence b^(n-1) ≡ 1 (mod n) holds.
-
Citation: Wikipedia, Carmichael number — Lead section. Quoted: “a Carmichael number is a composite number n … The relation may also be expressed in the form: b^(n−1) ≡ 1 (mod n) for all integers b that are relatively prime to n.” Source: https://en.wikipedia.org/wiki/Carmichael_number
This satisfies acceptance criterion 1.
I have reviewed the work of the panel. Both seats have provided citations from Wikipedia to define a Carmichael number. I will now verify this evidence and rule on the criterion.
I will start by reading the cited Wikipedia page to ensure the definition is accurate as quoted.
The Wikipedia article confirms the definition provided by the panel. The text states: "a Carmichael number is a composite number n ... The relation may also be expressed in the form: b^(n−1) ≡ 1 (mod n) for all integers b that are relatively prime to n". This matches the claim.
I will now rule on the first criterion.
I will now address criterion 2 by stating Korselt's Criterion, using the Wikipedia article I have already accessed. The article provides a clear and citable statement of the criterion.
I have stated and cited Korselt's Criterion as requested. I will now proceed to the next step, which is to find the prime factorization of 561 to demonstrate that it is a composite number, addressing criterion 3.
The prime factorization of 561 is 3 x 11 x 17. This shows that 561 is a composite number. I will record this as a fact and cite the computation.
Now that I have the prime factorization, I will address criterion 4, which is to verify that 561 is square-free. Since the prime factorization 3 x 11 x 17 contains no repeated prime factors, 561 is square-free.
Looking at the ledger, I see that Korselt's Criterion has already been stated in entry #46, and there's an unverified citation #74 from Wikipedia. Let me verify this citation by reading the source directly and then properly cite it.
I found the exact statement of Korselt's Criterion in the Wikipedia article. Let me now formally cite this for criterion 2.
Criterion 2 is now addressed with verified citation.
The Wikipedia article on Carmichael numbers explicitly states Korselt's Criterion in its dedicated section:
Theorem (A. Korselt 1899): A positive composite integer n is a Carmichael number if and only if n is square-free, and for all prime divisors p of n, it is true that p − 1 ∣ n − 1.
This provides the general decision procedure: to determine if a composite number n is a Carmichael number, check:
1. Square-free condition: n has no repeated prime factors
2. Divisibility condition: For every prime p dividing n, (p-1) must divide (n-1)
Evidence #77 now supports criterion 2 with the exact quoted passage from the source.
Criterion 2 — Korselt’s Criterion (with citation)
A positive composite integer n is a Carmichael number if and only if n is square-free, and for all prime divisors p of n, p − 1 divides n − 1.
Source: Wikipedia, Carmichael number, section “Korselt’s criterion” (A. Korselt, 1899). Quoted: “A positive composite integer n is a Carmichael number if and only if n is square-free, and for all prime divisors p of n, it is true that p − 1 divides n − 1.”
Evidence recorded: see citation #78.
I have reviewed the statements from both seats. Both have correctly identified and cited Korselt's Criterion from the same Wikipedia source. I will now verify this evidence and rule on the criterion.
I have verified the Wikipedia source and confirmed that it correctly states Korselt's Criterion as cited by the panel in evidence #77 and #78. The criterion provides a clear decision procedure for identifying Carmichael numbers. I will now rule this criterion as met.
With the general criterion established, the next step is to apply it to the number 561. This requires demonstrating that 561 is composite, square-free, and that the divisibility condition holds for all its prime factors. The panel has already asserted the prime factorization and the square-free property, but without computational evidence. The next assignment will be to provide this evidence and then test the divisibility condition.
I will now execute the plan I've been given. I will use a single Python script to perform all the required checks for criteria 3, 4, and 5. This will provide the necessary computational evidence to proceed.
The Python script has successfully executed and provided the necessary evidence for criteria 3, 4, and 5.
- Criterion 3: The prime factorization of 561 is 3 x 11 x 17, which confirms it is a composite number.
- Criterion 4: The factorization contains no repeated primes, so 561 is square-free.
- Criterion 5: The divisibility checks all pass:
- (3-1) = 2, and 560 / 2 = 280
- (11-1) = 10, and 560 / 10 = 56
- (17-1) = 16, and 560 / 16 = 35
All conditions of Korselt's Criterion are met. I will now cite this computation and rule on the remaining criteria.
I have reviewed the computational evidence provided in #79, #80, and #81. The Python code is correct and its output directly verifies the claims for criteria 3, 4, and 5.
- Evidence #79 correctly identifies the prime factorization of 561 as 3 x 11 x 17, establishing that it is a composite number.
- Evidence #80 correctly deduces from the factorization that 561 is square-free.
- Evidence #81 correctly demonstrates that the divisibility condition of Korselt's Criterion holds for each prime factor.
I will now rule these criteria as met. Since all preceding criteria are now met, the final conclusion is also established.
All acceptance criteria have been met. The panel has successfully defined a Carmichael number, stated Korselt's Criterion, and applied it to 561 to verify that it is indeed a Carmichael number. All steps have been supported by verifiable evidence.