561

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Summary

Factorization

561 is a square-free number with prime factors 3, 11, and 17. The prime factorization is:

$\!561=3*11*17$

Properties and families

Property or family	Parameter values	First few numbers satisfying the property	Proof of satisfaction/membership/containment
Carmichael number (also called absolute pseudoprime)	first Carmichael number	561, 1105, 1729, 2465, 2821, 6601, [SHOW MORE] 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461 View list on OEIS	The universal exponent is $\operatorname {lcm} \{3-1,11-1,17-1\}=80$ which divides $561-1=560$
Poulet number (Fermat pseudoprime to base 2)	second Poulet number	341, 561, 645, 1105, 1387, 1729, 1905, 2047, [SHOW MORE] 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341 View list on OEIS	follows from its being an odd Carmichael number.

Arithmetic functions

Function	Value	Explanation
Euler totient function	320	The Euler totient function is $(3-1)(11-1)(17-1)=(2)(10)(16)=320$.
universal exponent	80	The universal exponent is the least common multiple of $\{3-1,11-1,17-1\}$, which is $\operatorname {lcm} \{2,10,16\}$ and equals 80. Note that 561 is a Carmichael number precisely because the universal exponent divides 561 - 1 = 560.
Mobius function	-1	The number is a square-free number and it has an odd number of prime divisors (3 prime divisors).
divisor count function	8	$(1+1)(1+1)(1+1)$ where the first 1s in each sum denote the multiplicities of the prime divisors.
largest prime divisor	17
largest prime power divisor	17

Structure of integers mod 561

Abstract structure

The discussion here about the multiplicative group of integers modulo n relies on some background facts from group theory.
FACTS ABOUT MULTIPLICATIVE GROUPS: Classification of natural numbers for which the multiplicative group is cyclic | Characterization of multiplicative group of integers modulo n
GENERAL FACTS ABOUT GROUPS: order of direct product is product of orders, exponent of direct product is lcm of exponents

By the Chinese remainder theorem, we have the ring decomposition:

$\mathbb {Z} /561\mathbb {Z} \cong \mathbb {Z} /3\mathbb {Z} \times \mathbb {Z} /11\mathbb {Z} \times \mathbb {Z} /17\mathbb {Z}$

For the multiplicative group, this gives:

$(\mathbb {Z} /561\mathbb {Z} )^{*}\cong (\mathbb {Z} /3\mathbb {Z} )^{*}\times (\mathbb {Z} /11\mathbb {Z} )^{*}\times (\mathbb {Z} /17\mathbb {Z} )^{*}$

All groups on the right side are cyclic, and we get:

$(\mathbb {Z} /561\mathbb {Z} )^{*}\cong \mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /10\mathbb {Z} \times \mathbb {Z} /16\mathbb {Z}$

This group has order $2\times 10\times 16=320$ (the order of the multiplicative group is termed the Euler totient function and denoted $\varphi$, so this is saying that $\varphi (561)=320$).

The exponents of the factor groups are 2, 10, and 16 respectively, hence the exponent of the whole group is the lcm of these, which is 80. The exponent of the multiplicative group is termed the universal exponent and denoted $\lambda$, so this is saying that $\lambda (561)=80$).

Primality testing

Summary

Primality test	Relies on ...	Total number of candidate bases	Number of bases wih gcd more than 1	Number of witnesses that are relatively prime	Number of liars that are relatively prime	Total number of witnesses	Probability of picking witness	Probability of picking liar	Conditional probability of picking witness for relatively prime	Conditional probability of picking liar for relatively prime
Fermat primality test	Fermat's little theorem	560	240	0	320	240	240/560 = 3/7	4/7	0	1
Solovay-Strassen primality test	Euler's criterion	560	240	?	?	?	?	?	?	?
Miller-Rabin primality test		560	240	?	?	?	?	?	?	?

Fermat primality test

Further information: Fermat primality test

561 is a Fermat pseudoprime to base $a$ (or equivalently, $a$ is a Fermat liar for $n$) if the order of $a$ mod 3 divides 560, the order of $a$ mod 11 divides 560, and the order of $a$ mod 17 divides 560. Each of these conditions holds for all $a$ relatively prime to 561, because each of the numbers 3-1,11-1, 17-1 divides 560. Another way of putting this is that the lcm of these, which is the universal exponent, divides 560.

Thus, 561 is a Fermat pseudoprime to every base relatively prime to it (or equivalently, every number relatively prime to 561 is a Fermat liar for 561). Numbers with this property are termed Carmichael numbers or absolute pseudoprimes. Thus, 561 is a Carmichael number. In fact, it is the smallest Carmichael number.

This means that the probability of success for a single iteration of the Fermat primality test is the probability that a randomly chosen element from 1 to 560 has a common factor with 561. This is given by:

${\frac {561-\varphi (561)-1}{561-1}}={\frac {240}{560}}={\frac {3}{7}}$