# 101

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## 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 $7$th of $29$ primes of the form $2 x^2 + 29$:
$2 \times 6^2 + 29 = 101$ (Previous  ... Next)

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

No further terms of this sequence are documented on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## 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 $101$!