# 91

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## Number

$91$ (ninety-one) is:

$7 \times 13$

The $1$st integer whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
$91^2 = 8281$

The $1$st integer whose square has a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers:
$91^2 = 8281$

The $1$st Fermat pseudoprime to base $3$:
$3^{91} \equiv 3 \pmod {91}$

The $2$nd triangular number after $28$ which is the sum of $2$ cubes:
$91 = 27 + 64 = 3^3 + 4^3$

The $2$nd of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$

The $3$rd Fermat pseudoprime to base $4$ after $15$, $85$:
$4^{91} \equiv 4 \pmod {91}$

The $3$rd after $1$, $55$ of the $4$ square pyramidal numbers which are also triangular.

The $5$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
$10$, $34$, $58$, $85$, $91$, $\ldots$

The $6$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$:
$91 = 1 + 6 + 12 + 18 + 24 + 30 = 6^3 - 5^3$

The $6$th square pyramidal number after $1$, $5$, $14$, $30$, $55$:
$91 = 1 + 4 + 9 + 16 + 25 + 36 = \dfrac {6 \paren {6 + 1} \paren {2 \times 6 + 1} } 6$

The $7$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$:
$91 = 1 + 5 + 9 + 13 + 17 + 21 + 25 = 7 \paren {2 \times 7 - 1}$

The $8$th positive integer which cannot be expressed as the sum of a square and a prime:
$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $\ldots$

The $13$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$:
$91 = \displaystyle \sum_{k \mathop = 1}^{13} k = \dfrac {13 \times \paren {13 + 1} } 2$

The $17$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$:
$91 \to 9^2 + 1^2 = 81 + 1 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$

The $17$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$, $73$, $79$, $85$ such that no factorial of an integer can end with $n$ zeroes

The $30$th semiprime:
$91 = 7 \times 13$

## Historical Note

There are $91$ days in a quarter-year: $13$ weeks of $7$ days each.