17

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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, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, [SHOW MORE] 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 View list on OEIS	A natural number $n>1$ is prime if and only if $n$ is not divisible by any prime less than or equal to ${\sqrt {n}}$. Since ${\sqrt {13}}$ is between 4 and 5, we only need to check divisibility by primes less than or equal to 4, i.e., we need to verify that 13 is not divisible by the primes 2 and 3.
Fermat number, Fermat prime	$F_{2}$, where $F_{n}=2^{2^{n}}+1$, starts $n=0$	3, 5, 17, 257, 65537, 4294967297 View list on OEIS	plug and check
Proth prime: prime of the form $k\cdot 2^{n}+1$ with $2^{n}>k$	$n=4,k=1$	3, 5, 13, 17, 41, 97, 113, [SHOW MORE] 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857, 10369, 10753, 11393, 11777, 12161, 12289, 13313 View list on OEIS
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
near-square prime of the form $n^{2}+1$	$n=4$ (third prime of this form)	2, 5, 17, 37, 101, 197, 257, [SHOW MORE] 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177 View list on OEIS
lucky number of Euler	fifth of six such numbers	2, 3, 5, 11, 17, 41 View list on OEIS	A prime $p$ is a lucky number of Euler iff the ring of integers in $\mathbb {Q} ({\sqrt {1-4p}})$ is a unique factorization domain.

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
--	--	--	8 Yes, 8 No $8=\varphi (16)$, $\varphi$ is Euler totient function	8 residues, 8 nonresidues. Equal numbers of both.

Primitive root theory

Primitive roots

The number of primitive roots equals the number of generators of the additive group of integers mod 16, which is the Euler totient function of 16, which is 8. Given any primitive root $a$, the primitive roots are $a,a^{3},a^{5},\dots ,a^{15}$, i.e., the odd powers of $a$. 17 is a Fermat prime so the primitive roots are precisely the quadratic nonresidues, see quadratic nonresidue equals primitive root for Fermat prime.

The explicit list of primitive roots is: 3,5,6,7,10,11,12,14.

We note the following:

• The fact that 3 is a primitive root follows from the fact that Fermat prime greater than three implies three is primitive root.

Significance of 10 being a primitive root

If 10 is a primitive root modulo a prime $p$, then the prime $p$ is a full reptend prime in base 10, i.e., the decimal expansion of $1/p$ has a repeating block of the maximum possible length $p-1$. The condition holds for $p=17$, and the corresponding decimal expansion of 1/17 is:

$0\cdot {\overline {0588235294117647}}$

The corresponding number:

$\!0588235294117647$

has the property that it is a cyclic number, i.e., its first few multiples are obtained by cyclic permutations of its digits.

Quadratic theory

Quadratic residues and nonresidues

As noted above, the quadratic nonresidues coincide with the primitive roots and the quadratic residues are the remaining elements.

The quadratic residues are: 1,2,4,8,9,13,15,16.

The quadratic nonresidues are: 3,5,6,7,10,11,12,14.

Adjacent quadratic residues

Further information: Formula for number of pairs of adjacent quadratic residues modulo a prime

For a prime $p$ that is congruent to 1 mod 4, the number of pairs of adjacent squares mod $p$ is $(p+3)/4$ and the number of adjacent (both nonzero) quadratic residues is $(p-5)/4$. For $p=17$, these numbers are respectively 5 and 3.

The 3 pairs of adjacent quadratic residues for 17 are: (1,2), (8,9), and (-2,-1) (which is the same as (15,16).

If we allow for 0 as one of the pair members, the 5 pairs of adjacent squares are: (0,1), (1,2), (8,9), and (-2,-1), and (-1,0).

Paley graph and Ramsey theory

The structure of quadratic residues are nonresidues for the prime 17 has a combinatorial significance. Specifically, consider the Paley graph of integers mod 17, defined as the graph on the integers mod 17 where two vertices are connected if their difference is a quadratic residue mod 17. This is a self-complementary graph and it does not have a clique of size four, hence it can be used to show that the Ramsey number $R(4,4)$ is at least 18.

Condition for 17 to be a quadratic residue modulo a prime

Whether or not 17 is a quadratic residue modulo a prime depends only on the congruence class of the prime modulo 68 (obtainedas 4 times 17). This follows from quadratic reciprocity for odd primes.

By Dirichlet's theorem on primes in arithmetic progressions, we thus find that there are infinitely many primes $p$ for which 17 is a quadratic residue and infinitely many primes $p$ for which 17 is a quadratic nonresidue.

Square roots of -1

Since $17\equiv 1{\pmod {4}}$, -1 has a square root mod 17. Its two square roots mod 17 are 4 and -4 (which is 13 mod 17).

We can also see this from the fact that $17=F_{2}=2^{2^{2}}+1$, and that the square roots of -1 mod $F_{n}$ are $\pm 2^{2^{n-1}}$.

Waring representations

Sums of squares

Item	Value
unique (up to plus/minus and ordering) representation as sum of two squares	$4^{2}+1^{2}$. Note that existence and uniqueness both follow from it being a prime that is 1 mod 4. This also corresponds to the factorization $17=(4+i)(4-i)$ in the ring of Gaussian integers $\mathbb {Z} [i]$.
representations as sum of three squares (up to ordering and plus/minus equivalence)	$3^{2}+2^{2}+2^{2}$ $4^{2}+1^{2}+0^{2}$

Sums of cubes

Item	Value
representation as sum of two cubes	not possible, because $A^{3}+B^{3}=(A+B)(A^{2}-AB+B^{2})$ is not possible
representation as sum of three cubes	$2^{3}+2^{3}+1^{3}$ is the only one using nonnegative inputs.
representation as sum of four cubes	none using nonnegative inputs. Using both positive and negative inputs: $3^{3}+(-2)^{3}+(-1)^{3}+(-1)^{3}$

Sums of fourth powers

Item	Value
representation as sum of two fourth powers	$2^{4}+1^{4}$

Cyclotomic theory

Cyclotomic extension of primitive roots of unity

Fill this in later

Constructibility of 17-gon

17 is a Fermat prime, hence the regular 17-gon is constructible by straightedge and compass. This is because the cyclotomic extension of adjoining 17th roots of unity can be obtained by taking successive quadratic extensions.

Polynomials

Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 17 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 17
$n^{2}-n+5$	2	all numbers 1,2,3,4, because 5 is one of the lucky numbers of Euler.	4
$n^{2}-n+11$	2	all numbers 1-10, because 11 is one of the lucky numbers of Euler.	3
$n^{2}-n+17$	2	all numbers 1-16, because 17 is one of the lucky numbers of Euler.	1

Irreducible polynomials by Cohn's irreducibility criterion

By Cohn's irreducibility criterion, we know that if we write 17 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here all the irreducible polynomials:

Base $b$	17 in base $b$	Corresponding irreducible polynomial
2	10001	$x^{4}+1$
3	122	$x^{2}+2x+2$
4	101	$x^{2}+1$
5	32	$3x+2$
6	25	$2x+5$
7	23	$2x+3$
8	21	$2x+1$
9	18	$x+8$
10	17	$x+7$
11	16	$x+6$
12	15	$x+5$
13	14	$x+4$
14	13	$x+3$
15	12	$x+2$
16	11	$x+1$

Multiples

Interesting multiples

Number	Prime factorization	What's interesting about it
561	3 times 11 times 17	smallest Carmichael number, i.e., smallest absolute pseudoprime
1105	5 times 13 times 17	second Carmichael number, i.e., second absolute pseudoprime
2465	5 times 17 times 29	fourth Carmichael number, i.e., fourth absolute pseudoprime