17

Summary

Factorization

The number 17 is a prime number.

Properties and families

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	it is the 7th prime number	2,3,5,7,11,13,17,19,23,29,31, ... (never stops, infinitude of primes)	divide and check
Fermat number, Fermat prime	$F_2$, where $F_n=2^{2^n}+1$, starts $n=0$	3,5,17,257,65537	plug and check
regular prime	sixth regular prime (2 is neither regular nor irregular)	3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE] 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431 View list on OEIS

Structure of integers mod 17

Discrete logarithm

We can take 3 to be a primitive root mod 17, i.e., a generator for the multiplicative group of integers mod 17. With this, the discrete logarithm table from the multiplicative group mod 17 to the additive group mod 16 looks as follows:

Congruence class mod 17 (written as smallest positive integer)	Congruence class mod 17 (written as smallest magnitude integer)	Discrete logarithm to base 3, written as integer mod 16	Is it a primitive root mod 17 (if and only if the discrete log is relatively prime to 16)?	Is it a quadratic residue or nonresidue mod 17 (residue if discrete log is even, nonresidue if odd)?
1	1	0	No	quadratic residue
2	2	14	No	quadratic residue
3	3	1	Yes	quadratic nonresidue
4	4	12	No	quadratic residue
5	5	5	Yes	quadratic nonresidue
6	6	15	Yes	quadratic nonresidue
7	7	11	Yes	quadratic nonresidue
8	8	10	No	quadratic residue
9	-8	2	No	quadratic residue
10	-7	3	Yes	quadratic nonresidue
11	-6	7	Yes	quadratic nonresidue
12	-5	13	Yes	quadratic nonresidue
13	-4	4	No	quadratic residue
14	-3	9	Yes	quadratic nonresidue
15	-2	6	No	quadratic residue
16	-1	8	No	quadratic residue