# 13

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## Number

$13$ (**thirteen**) is:

- The $6$th prime number, after $2$, $3$, $5$, $7$, $11$

- The $1$st power of $13$ after the zeroth $1$:
- $13 = 13^1$

- The $1$st emirp:
- $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $\ldots$

- The larger of the $3$rd pair of twin primes, with $11$

- The $2$nd Pythagorean prime after $5$, and so by Fermat's Two Squares Theorem the sum of two squares uniquely:
- $13 = 4 \times 3 + 1 = 4 + 9 = 2^2 + 3^2$

- The $2$nd prime number $p$ after $3$ the period of whose reciprocal is $\dfrac {p - 1} 2$:
- $\dfrac 1 {13} = 0 \cdot 076923 \, 076923 \ldots$

- The $2$nd integer after $7$ the decimal representation of whose square can be split into two parts which are each themselves square:
- $13^2 = 169$; $16 = 4^2, 9 = 3^2$

- The $2$nd Wilson prime after $5$:
- $13^2 \divides \paren {13 - 1}! + 1 = 479 \, 001 \, 601$

- The $2$nd after $4$ in the sequence formed by adding the squares of the first $n$ primes:
- $13 = \ds \sum_{i \mathop = 1}^2 {p_i}^2 = 2^2 + 3^2$

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

- The $4$th Fibonacci prime after $2$, $3$, $5$.

- The $4$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $3$, $5$, $11$:
- $13 \# - 1 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 - 1 = 30 \, 029$

- The $4$th happy number after $1$, $7$, $10$:
- $13 \to 1^2 + 3^2 = 1 + 9 = 10 \to 1^2 + 0^2 = 1$

- The index of the $5$th Mersenne prime after $2$, $3$, $5$, $7$:
- $M_{13} = 2^{13} - 1 = 8191$

- The $5$th lucky number:
- $1$, $3$, $7$, $9$, $13$, $\ldots$

- The $6$th permutable prime after $2$, $3$, $5$, $7$, $11$.

- The index of the $6$th Mersenne number after $1$, $2$, $3$, $5$, $7$ which Marin Mersenne asserted to be prime

- The $7$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$:
- $13 = 5 + 8$

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

- The $8$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$:
- $13 = 2 + 11$

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

- The $11$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

- The $11$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ such that $2^n$ contains no zero in its decimal representation:
- $2^{13} = 8192$

- The hypotenuse of the $5-12-13$ Pythagorean triangle.

- The square of the reverse of $13$ equals the reverse of the square of $13$:
- $13^2 = 169$
- $31^2 = 961$

- With $x = 239$, the only $y$ which is the solution of the indeterminate Diophantine equation $x^2 + 1 = 2 y^4$:
- $239^2 + 1 = 2 \times 13^4$

## Also see

- Recurring Parts of Multiples of One Thirteenth
- Twelve Factorial plus One is divisible by 13 Squared
- Square of Reversal of Small-Digit Number
- Solution of Diophantine Equation $x^2 + 1 = 2 y^4$

*Previous ... Next*: Proth Prime*Previous ... Next*: Pythagorean Prime*Previous ... Next*: Fibonacci Prime*Previous ... Next*: Wilson Prime

*Previous ... Next*: Index of Mersenne Prime*Previous ... Next*: Mersenne Prime/Historical Note*Previous ... Next*: Squares whose Digits can be Separated into 2 other Squares

*Previous ... Next*: Fibonacci Number

*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Lucky Number

*Previous ... Next*: Happy Number

*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares*Previous ... Next*: Ulam Number*Previous ... Next*: Prime Number*Previous ... Next*: Twin Primes*Previous ... Next*: Permutable Prime*Previous ... Next*: Sequence of Prime Primorial minus 1

*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes

## Historical Note

The number $13$ is traditionally unlucky.

This is apparently based on the fact that there were $13$ people who attended the Last Supper.

It is believed that this superstition originated in the middle ages.

The word **triskaidekaphobia** means **fear of the number $13$.**

The fact that $13$ is in fact classified by number theorists as a lucky number is just one more indication of how mathematicians delight in confusing muggles.

Apart from that, there are several instances of the number $13$ in the mundane world:

- There are $13$ cards in each of the $4$ suits of a deck of cards

- The number of objects in a baker's dozen.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $13$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $13$

Categories:

- Powers of 13/Examples
- Proth Primes/Examples
- Fibonacci Primes/Examples
- Wilson Primes/Examples
- Indices of Mersenne Primes/Examples
- Mersenne's Assertion/Examples
- Fibonacci Numbers/Examples
- Lucky Numbers/Examples
- Happy Numbers/Examples
- Ulam Numbers/Examples
- Prime Numbers/Examples
- Twin Primes/Examples
- Permutable Primes/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Emirps/Examples
- Specific Numbers
- 13