# 17

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## Number

$17$ (**seventeen**) is:

- The $7$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$

- The $1$st integer to be the sum of $2$ distinct powers of $4$:
- $17 = 1 + 16 = 1^4 + 2^4$

- The $1$st odd positive integer not expressible in the form $2 n^2 + p$ where $p$ is prime

- The $2$nd emirp after $13$

- The $2$nd long period prime after $7$:
- $\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$

- The $3$rd Stern number after $1$, $3$

- The $3$rd Stern prime after $1$, $3$

- The $3$rd Fermat number and Fermat prime after $3$, $5$:
- $17 = 2^{2^2} + 1$

- The $3$rd positive integer $n$ after $5$, $11$ such that no factorial of an integer can end with $n$ zeroes

- The $3$rd prime number of the form $n^2 + 1$ after $2$, $5$:
- $17 = 4^2 + 1$

- The $3$rd integer after $2$, $5$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied

- The $4$th Proth prime after $3$, $5$, $13$:
- $17 = 1 \times 2^4 + 1$

- The $4$th Dudeney number after $0$, $1$, $8$, and only prime Dudeney number:
- $17 = 4 + 9 + 1 + 3$, while $17^3 = 4913$

- The smaller of the $4$th pair of twin primes, with $19$

- The $5$th of the lucky numbers of Euler after $2$, $3$, $5$, $11$:
- $n^2 + n + 17$ is prime for $0 \le n < 15$

- The index of the $6$th Mersenne prime after $2$, $3$, $5$, $7$, $13$:
- $M_{17} = 2^{17} - 1 = 131 \, 071$

- The $6$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$

- The $7$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$

- The $7$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
- $3$, $4$, $9$, $10$, $12$, $16$, $17$, $\ldots$

- The index of the $7$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$ which Marin Mersenne asserted to be prime

- The $9$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
- $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $\ldots$

- The $12$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

- The $12$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$ such that $5^n$ contains no zero in its decimal representation:
- $5^{17} = 762 \, 939 \, 453 \, 125$

- There are $17$ wallpaper groups

## Also see

- Construction of Regular Heptadecagon
- Prime Dudeney Number
- Smallest Odd Number not of form 2 a squared plus p
- Reciprocal of 17
- 17 Wallpaper Groups

*Previous ... Next*: Stern Number*Previous ... Next*: Stern Prime

*Previous ... Next*: Primes of Form n^2 + 1*Previous ... Next*: Prime Number Race between 4n+1 and 4n-1*Previous ... Next*: Fermat Number*Previous ... Next*: Fermat Prime

*Previous ... Next*: Dudeney Number

*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Numbers of Zeroes that Factorial does not end with*Previous ... Next*: Euler Lucky Number

*Previous ... Next*: Prime Number*Previous ... Next*: Twin Primes*Previous ... Next*: Index of Mersenne Prime*Previous ... Next*: Mersenne Prime/Historical Note*Previous ... Next*: Left-Truncatable Prime*Previous ... Next*: Pythagorean Prime*Previous ... Next*: Emirp*Previous ... Next*: Permutable Prime*Previous ... Next*: Proth Prime

*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: 91 is Pseudoprime to 35 Bases less than 91

## Historical Note

$17$ was the highest number whose square root was proved irrational by Theodorus of Cyrene.

Plutarch reports that the Pythagoreans had a horror of the number $17$.

That was because it lay half way between $16$ and $18$, which are the areas of rectangles whose areas equal their perimeters.

$17$ seems to be a popular age to write a pop song about:

to name but three.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 13$: The fundamental theorem of arithmetic - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $17$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $17$

Categories:

- Stern Numbers/Examples
- Stern Primes/Examples
- Prime Number Races/Examples
- Fermat Numbers/Examples
- Fermat Primes/Examples
- Long Period Primes/Examples
- Dudeney Numbers/Examples
- Euler Lucky Numbers/Examples
- Prime Numbers/Examples
- Twin Primes/Examples
- Indices of Mersenne Primes/Examples
- Mersenne's Assertion/Examples
- Left-Truncatable Primes/Examples
- Emirps/Examples
- Permutable Primes/Examples
- Proth Primes/Examples
- Specific Numbers
- 17