# 101

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## Contents

## Number

$101$ (**one hundred and one**) is:

- The $26$th prime number

- The $1$st of the $9$th pair of twin primes, with $103$

- The $3$rd prime number after $5$, $19$ of the form $\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
- $101 = 5! - 4! + 3! - 2! + 1!$

- The upper end of the $5$th record-breaking gap between twin primes:
- $101 - 73 = 28$

- The $5$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$:
- $101 = 10^2 + 1$

- The $5$th positive integer after $1$, $2$, $7$, $11$ whose cube is palindromic:
- $101^3 = 1 \, 030 \, 301$

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

- The $7$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$ whose square is also palindromic integer
- $101^2 = 10 \, 201$

- The $12$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$ such that the Fibonacci number $F_n$ ends in $n$

- The $36$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
- $1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $101$, $\ldots$

- The $50$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$, $100$ which cannot be expressed as the sum of distinct pentagonal numbers

## Also see

*Previous ... Next*: Sequence of Integers whose Cube is Palindromic*Previous ... Next*: Palindromic Prime*Previous ... Next*: Prime Factors of One More than Power of 10

*Previous ... Next*: Prime Number*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares

## Historical Note

The number $101$ has several cultural significances.

- The rhetorical device meaning:
**a large number ($100$) and then some (and $1$)**, often in the context of book titles:

*The Hundred and One Dalmatians*, and*101 Uses for a Dead Cat*, and so on.

**Room $101$**: In*Nineteen Eighty-Four*by George Orwell,**room $101$**is the name of the torture chamber where bad things happen. It has entered contemporary consciousness for a rhetorical and metaphorical space into which things are to be consigned which are particularly hated.

- As a metaphor for an entry-level course of study. It is usual, in Western universities, for the first course in the first year of study of a degree, to be provided with the serial number
**xx$101$**, where**xx**identifies the nature of the study course. Hence it is used rhetorically, often for the purposes of ridicule, to mean**basic information which you really ought to know by now**.

*Oh come on, forgetting to set the timer? That's cooking $101$!*

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $101$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $101$