341

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Summary

Factorization

The factorization is as follows, with factors 11 and 31:

$341=11*31$

Properties and families

Property or family	Parameter values	First few members of the family	Proof of satisfaction/membership/containment
Poulet number (also called Sarrus number), i.e., Fermat pseudoprime to base 2	smallest 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	$2^{10}=1024\equiv 1{\pmod {341}}$, so $2^{340}\equiv 1{\pmod {341}}$.

Arithmetic functions

Function	Value	Explanation
Euler totient function	300	The Euler totient function is $(11-1)(31-1)=(10)(30)=300$.
universal exponent	30	The universal exponent is $\operatorname {lcm} \{10,30\}=30$.
divisor count function	4	$(1+1)(1+1)$ where the first 1s in both factors are the multiplicities of the prime divisors.
divisor sum function	384	$(11^{2}-1)/(11-1)$ times $(31^{2}-1)/(31-1)$ equals $(12)(32)=384$.
Mobius function	1	The number is square-free and has an even number of prime divisors (2 prime divisors).

Structure of integers mod 341

Abstract structure

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

$\mathbb {Z} /341\mathbb {Z} \cong \mathbb {Z} /11\mathbb {Z} \times \mathbb {Z} /31\mathbb {Z}$

For the multiplicative group, this gives:

$(\mathbb {Z} /341\mathbb {Z} )^{*}\cong (\mathbb {Z} /11\mathbb {Z} )^{*}\times (\mathbb {Z} /31\mathbb {Z} )^{*}$

Both groups on the right side are cyclic with orders 10 and 30 respectively, so the whole group has order $10\times 30=300$ (the order of the multiplicative group is termed the Euler totient function and denoted $\varphi$, so this is saying that $\varphi (341)=300$).

The exponents of the factor groups are 10 and 30 respectively, hence the exponent of the whole group is the lcm of these, which is 30. The exponent of the multiplicative group is termed the universal exponent and denoted $\lambda$, so this is saying that $\lambda (341)=30$).

Primality testing

Fermat test

A base $a$ is a Fermat pseudoprime mod 341 if the order of $a$ mod 11 divides 340 and the order of $a$ mod 31 divides 340. The first condition always holds, because 11 - 1 = 10 divides 340. The second condition is equivalent to the order dividing 10, which holds for 1/3 of the bases relatively prime to 31. Overall, 341 is a Fermat pseudoprime for 1/3 of the possible bases relatively prime to it. This means that if we pick use the Fermat test for a base picked at random, there is a 2/3 chance that we will be able to use it to conclude that 341 is not prime.

2 is one of the bases for which 341 is a Fermat pseudoprime, because $2^{10}=1024\equiv 1{\pmod {341}}$. Thus, 341 is a Poulet number or Sarrus number.

Euler test

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Strong Fermat test

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